After more than 30 years, the Dove ice cream bar is an overnight sensation. Following decades when the bell-shaped bars with the thick bittersweet chocolate coating were known only to those who patronized a certain candy and ice cream store on Chicago's South Side, DoveBars (as they are now called) have gone coast to coast.

People Magazine has done a feature. The Today Show visited Burr Ridge, Ill., and filmed a segment. Paul Harvey has called. And public demand has been so explosive that Dove's new factory is running three shifts around the clock, six days a week. There is a backlog of orders, no new sales territories are being opened and the bars are being allocated to existing customers. The tight supply and overwhelming demand has led one newspaper to dub the DoveBar "the Cabbage Patch" of ice cream.

DoveBars come in five ice cream flavors (vanilla, chocolate, strawberry, coffee and coconut) and, as good as the ice cream is, many fans consider it just an excuse for indulging in the rich chocolate coating. "So sinful, they're wonderful," wrote the Chicago Sun-Times.

The large bars, which are six ounces each, are being marketed as "an adult superpremium dessert." At $2-plus each when sold from pushcarts on the streets of Manhattan and in other cities, and $2.99 for a package of two in a supermarket, the bars belong in the upscale niche occupied by Haagen-Dazs, Frusen Gladje and other top-quality frozen treats.

But this product didn't spring instantly from a Madison Avenue demographic survey. It has been made lovingly for decades at the back of Dove Candies, a Chicago soda fountain and confectionary.

DoveBars got their start in the 1950s, the solution to candy maker Leo Stefanos' irritation at his young sons chasing after ice cream trucks. The elder Stefanos, a Greek immigrant, was proud of the ice cream and candy he had made and sold in his shop since 1939, and it irked him to see his two boys running after competitors.

So, in a little ceremony that younger son Mike, now 33, remembers well, Leo Stefanos gathered his family around and hand-dipped the first bars. He intended them for family consumption only, but within two weeks he was selling them in his store. There they remained, made much the same way for decades.

Homemade ice cream was frozen overnight in rectangular paper cartons. In the morning, an employe wielding a large knife would cut the ice cream blocks into slabs, insert sticks and dip the bars into a bowl of warm chocolate.

The ingredients, including Dutch-processed chocolate, are listed on the bars' foil wrapper, but the exact recipe is secret. "There's a lot of things in the coating," said Mike Stefanos, now president of the company. "All the things came about from our burning our hands in chocolate all these years."

By 1980, DoveBars were being delivered to a few Chicago-area drug stores, groceries and country clubs. Production was 500 bars a day. In order to sign on with the larger supermarkets, a distributor was found, and by 1982, production had risen to about 1,000 bars a day. Currently, with eight employes hand-dipping on the production line at all times, a staggering 72,000 bars a day are turned out.

DoveBars was able to make this huge jump with the help of two financial angels, brothers Jim and Dick Zacharias of Chicago. The brothers, ages 73 and 71, respectively, are in the business of plating precious metals but also have numerous other investments.

DoveBars came to Dick Zacharias' attention several years ago when his wife, knowing his love of chocolate, urged him to sample one at their country club. Zacharias went to see Mike Stefanos about expanding the operation, but Stephanos was hesitant. He didn't think it was possible to maintain the quality of the bars on a large scale.

For the Stefanos family, there was a lot more than money at stake in an expansion. When Leo Stefanos died in 1977 at the age of 74, his wife Sophia, sons Mike and Chris and daughter Amy continued the business exactly as he left it.

"It was a way to keep Dad's memory alive," said Mike Stefanos. It took Dick Zacharias two years to persuade the family that he wanted to keep the bar exactly as it was -- just make more of them.

In January 1984, the Zachariases brought in Lou Yaseen, who had been a financial consultant for their business interests, as president. That June, DoveBars had a wildly enthusiastic debut at the Fancy Food Show in Washington; since then, life has been hectic for the Zacharias and Stefanos families and Lou Yaseen.

The DoveBar's rise has been so rapid that even company officials are shaking their heads. In March of this year, production moved from the back of the confectionary to a factory in an industrial park in Burr Ridge, a southwestern suburb. At that time, there were no customers outside the Chicago area.

Within three months, DoveBars were being shipped to 3,500 supermarkets around the country. They now are sold in 24 states and the District of Columbia, with other states and Canada on the waiting list. Even the most optimistic projections have proved woefully low.

"We have been so readily accepted," marveled Yaseen, 40. "Instead of being in the top 200 stores in a market, suddenly we're in the top 1,500. In a store where a typical, cheaper ice cream bar might sell two cases a week and we optimistically thought we might sell three, there are situations where we are selling 20 to 30 cases a week."

A recent sales estimate of $10 million for 1985 is being revised "substantially upward."

Although the company is straining at the seams to meet demand, Yaseen says it is not bedeviled by undercapitalization -- the bane of many a firm enjoying sudden popularity -- and will be adding $2 million in plant improvements by the end of the year.

Mike Stefanos says the collaboration of the two families has been "fantastic. I view it as a family extension."

Yaseen notes that he and Mike Stefanos have practically been living together for the past year and a half, but that nary a harsh word has passed between them.

For the Stefanos family, there is a delicate juggling act going on, as they balance the need to maintain the Dove quality on a national scale and at the same time preserve a traditional presence in the neighborhood shops on South Pulaski Road and in the southern suburbs of Oak Lawn. Both shops feature hand-made chocolates, 10 flavors of homemade ice cream and superlative fountain treats -- all of which take time and skill to produce.

Currently, Mike Stephanos is meeting both responsibilities by working from 5:30 to 8:30 a.m. at the Pulaski shop, then spending the rest of the day at the plant. Saturdays he spends at the shop, concentrating on candy-making.

His sister, who runs the store in Oak Lawn, and his mother, who is at the Pulaski location, are having to work harder than ever, he says.

The family is keenly aware of neighborhood pressure to remain unchanged. "The neighborhood customers come in and say, 'Don't you leave us. Don't you change,' " he says.

Some change is already in the works, however. Haagen-Dazs has announced plans to produce an ice cream novelty soon, and Dove will be introducing national product No. 2 next spring. President Yaseen won't describe it, other than to say it will contain ice cream and chocolate.

The product has been dreamed up by Mike Stefanos, using the lessons he learned from his father. "That's what we've got over the other ice cream companies -- the candy-making experience," notes Yaseen.

Mike Stefanos says he is even happier with the quality of the DoveBar now than when they were made in the back of the confectionary, thanks to new freezers that chill faster and colder, resulting in smoother ice cream. He is pleased about the warm relationship his family has forged with its financial backers and is delighted by all the favorable public response.

But the biggest thrill is seeing all the boxes of DoveBars ready for shipment with his father's dove of peace logo on the side. "My mom had that dove etched on his gravestone," said Stefanos, "and to keep it going means a lot. The best compliments I get are when people say, 'Your dad would have been proud.' "