Dick Cavett has run out of television networks but he has not run out on television. The public TV stations have met, voted, and elected him a talk-show host. Starting in the fall, he will be on public TV stations, including Channel 26 in Washington, nightly at 11 o'clock.

In the meantime, Cavett will be kept off the streets by starring in a Broadway comedy. He takes over the Tom Cortenay lead role in "Otherwise Engaged" starting July 21.

So there's more than one reprieve for all the people who were afraid they wouldn't have Dick Cavett to kick around any more. And, of course, for Dick Cavett.

There is one little thing blocking Cavett's return to TV: $800,000. Public TV stations have already put up $1 million and New York's Channel 13 is now looking for the other $800,000 to produce the show. They expect some high-minded and stout-hearted corporation to step forward at any moment.

How much money Cavett himself will clear is not certain, but it will not be a network-type fortune. "You don't do public TV for the money," says Cavett. "Besides, I hope to pass as a civilian and win the '$20,000 Pyramid.'"

Is it worth all this trouble to get Cavett back on television? Yes it is. Until TV is overrun with wits, we are going to need him and, yes, have to put up with his little fallibilities. It's encouraging that someone as smart as Cavett hasn't turned his back on television, even though it has repeatdly turned its back on him.

"The fact is, I like television," he says. "I'd like to remain in it and I enjoy doing it and I think I'm suited to it." Then he pulls on his hair. He will continue pulling on it for about 30 minutes, until it is standing on its heels and he can ask, "Do you like my hair better this way?" with half a Stan Laurel grin. It's a ridiculously rainy Manhattan afternoon, and Cavett has just burst into his 13th-floor office splattering water and sputtering expletives.

He still looks much closer to 30 than to 40. As it happens, though, he is 40.

The prospect of doing nightly, half-hour, one-on-one interviews and not having to fit them between commercials obviously pleases him. "Honestly, it's the best opportunity I've had in many ways, a chance to I've had in many ways, a chance to get the best people from anywhere - the arts, culture, science and politics - to sit and talk in an uninterrupted way. I don't mean it to be either ponderous on the one hand or just frothy on the other, but a show where you can have a Pulitzer Prize-winning author one night and Woody Allen the next."

What if Woody Allen wins the Pulitzer Prize? Oh, never mind.

"I don't want it to sound like a show that's 'good for you,' a morally improving experience," Cavett says. "Or just a silly-ass talk show either. But I think I know how to strike that balance."

Asked to name sample guests, Cavett huffs, "I refuse to play the name-dropping game, though earlier he had unashamedly bragged, "Baryshnikov and Bureyev both told me I should have been a dancer."

Cavett has been hanging around the smart arts set lately doing half-time interviews on public TV operas and ballets. It was his strong showing there that made his long-hoped-for public TV series possible. Even so, this business of starring in Broadway play does sound like over-reaching, although the man who will never let us forget he went to Yale did appear in many student productions there and in summer stock later.

"To say 'Dick Cavett is back on Broadway' would be stretching it, however," he concedes. "My only other appearances on Broadway were the Actors' Fund benefit speeches a few years ago. They'd get these poor, downtrodden, out-of-work actors to strike a pose onstage and do the Actors' Fund pitch.

"I was funny, yeah. I thought that was the way to do it. I got a lot of laughs. I thought, 'Someone will surely discover me doing this; I'm on my way, this is it, I'm on Broadway and I'm getting laughs.'

"But, no one notices."

Cavett admits that people drawn to see him in "Otherwise Engaged," in the role of a beleaguered intellectual fussbudget, will be mainly those who want to see a TV personality in person rather that witness one of the great acting jobs of the century.

"There is something slightly immoral about the fact that somebody who's known from television- what am I trying to say? - that someone who's simply a television personality can attract people to legitimate theater. I'm sure to some people it seems like an illegitimate route to the stage. I do hope to enjoy it, but I am taking it seriously. It isn't just a fling to indulge myself. I miss acting. There are certain satisfactions you get from it that you can't get from anything else.

"My wife [Carrie Nye] is a real actress. She has such high standards she hardly ever works. She says I'm madening to work with. I have this terrible habit of not knowing my lines until opening night, which is not the nicest thing to do to the other actors."

Doing a play is by no means "running away from television," and Cavett will indeed be back. While often praised by critics as the model inquisitor. Cavett has been bouncing around TV from one near-niche to another for years. In the process, he has had the chanve to be manhandled by all three networks, though he says he has no "particular" feelings about being "treated badly."

One could be cynical and call this his last chance to be abused. But public TV could be just the place for a guy who has the decency and the intelligence to wonder aloud if he is misusing the word "hopefully" as he talks.

"Hopefully, I'm not," Cavett says.

Taking it from the top down network memory lane, we find that ABC toyed with Cavett the longest. When one considers the sludge now aired by that network in the 11:30 p.m. slot, it's rather amazing to think that this 90-minutes nightly once belonged to Cavett and his little citadel of civility. His later life at ABC included a show programmed three times a week in prime time. "Groucho told me, 'You need a secretary to know when the show is on,'" Cavett recalls.

"I came to ABC at a time when we weren't exactly compatible. They didn't know what to do with me, nor I with them exactly. The fact that I remained there for six years is a tribute to both of us, really."

A ballyhooed deal with CBS to headline a talk-variety showcase produced only a few bleak specials and an off-killer summer series. "That was doomed from the start I think because I was brought in as Fred Silverman's baby and I became Fred Silverman's orphan because he left for ABC almost immediately after I got there. I did a summer show for four weeks that was mostly deplorable and misconceived from top to bottom." ut the best was yet to come: NBC, where Cavett got the entirely unenviable task of cohosting that slap-happy fiasco "The Big Party," which began the "Big Event" season on the network. Everything that could go wrong did until an exasperated Cavett was finally reduced to making hand shadows. Was this his most excruciating TV experience ever?

"Yes. Next question."

He is pressed for lurid details.

"Really, what is there to say? When you can see the floor manager throw his earphones on the floor and stamp on them because he can't get in touch with the control room, or when a guy frantically cues you toward a camera and there's no camera there, what can you do? That was one of those shows where you're doing live television and people say, 'Isn't it wonderful? Here we are, six hours before air time, and it's chaos, but tonight it'll all come together.'

"Not that time."

In spite of everything, Cavett has faith in TV. "I still like the medium for myself. The talk form I think does really well. You get the full impact of it as you don't quite from almost anything else that's seen on television. It may simply have to do with size: a face talking on television is about the same size as it is in real life, and it's compelling when it's the right person."

Cavett has shown a greater knack for snaring the right persons than any other TV talk host. True, it may be hard to twirl up great whelps of enthusiatic gush over him, but that's partly because he's so sane, sensible, cliche-conscious and calm. He's easy to like but hard to love.

Of course, not everybody can be Walter Cronkite, paterfamilias to a nation.

Still, Cavett sometimes seems as chilly as the frost-free Amana with the ice water in the door.

And then again, at least he doesn't try to grovel his way into our hearts.

All this may help explain why, when Newsday columnist Marvin Kittman started a "Dollars for Dick" fund to get Cavett onto public TV, he was only able to raise $6.50. Fortunately, the money came forth from the stations themselves.Would this heart have been broken if the stations hadn't voted him in? "I lost the student council election in ninth grade, so I'm no stranger to defeat," Cavett says.

"It's interesting - I wonder how the returns come in." Look out, he's thinking. "Do they have a map - 'Yes in New Jersey for Cavett, but down in the Corn Belt and the rustic plantations of the Old South, the returns are not yet in. And from the Halls of Montezuma -' Say, where are the Halls of Montezuma anyway? You never hear anything about the rooms. Just the halls."

He's a Yale man, folks. Forgive him.

At least Cavett doesn't pretend he hasn't missed being on television five nights a week, something he hasn't done since his ABC late-night show went off very early in 1975.

"Being on TV every night is something you don't think you could every miss while you are doing it," says Cavett, making one last yank at his hair. "But later, you can. When the pressure's gone, it's like being over a headache. It's like remembering a day when you had a terrific headache and you went to the circus, but you only remember the circus, and so it seems like it was fun."