American Telephone & Telegraph Co. has asked the government for permission to market a new televised-conferencing service that would begin operations between Washington and New York next March.

The service, to be known as "Picturephone Meeting Service," will be available in 16 cities next year and in a total of 42 cities by the end of the following year, the company said in documents filed last week with the Federal Communications Commission.

AT&T said the service would be available in public rooms in each of the 42 cities or through private teleconferencing rooms that AT&T is offering to construct on the premises of a corporation or other institution.

A one-hour conference between public rooms in Washington and New York would cost $1,340, while a meeting between private rooms served by AT&T in those two cities would cost $600.

Although customers could install their own facilities, the installation of private facilities through the Bell System would cost $124,800 in addition to monthly equipment rental and access line fees of $13,240. A monthly charge of $250 a mile would be added to connect each room to AT&T facilities.

The service will be in full color, and the sets could show slides and other graphics, copy images displayed on a screen and tape either the incoming or outgoing pictures.

The company announced last spring that it would replace its limited "Picturephone" system with the more sophisticated service described last week. That system had been offered in black and white to 12 cities on an "experimental" basis until the tariffs for the service expired last June, a company spokesman said.

Although the results from that system did not match the fanfare it attracted when introduced nearly 20 years ago, the high costs and other difficulties increasingly associated with travel make the system more attractive today. "Video teleconferencing increases productivity by speeding the corporate decision-making process," said John Wyman, vice president of marketing for AT&T Long Lines.

A number of large and small concerns have gotten into the business, providing similar services for hotel chains and other businesses. Satellite companies and other communications concerns also have been studying teleconferencing with an eye toward lowering the high cost of installing studio facilities.