The cornerstone of the new, long-delayed U.S. Embassy was officially set in the ground here today, marking the start of a new chancery to replace the present aging, overcrowded and much-bugged U.S. mission in central Moscow.

To be completed in December, 1983, the U.S.-designed complex of offices, living quarters and school should bring better security for the Americans as well as its inescapable byproduct -- even greater physical isolation from the Soviets than now exists for the U.S. diplomatic community.

Both Ambassador Malcolm Toon and Moscow Deputy Mayor N. S. Trofimov in formal remarks and toasts at a champagne reception on the muddy site after the grandlaying declared that the startup attests to the enduring relations between the two nations.

The estimated $100 million new array of buildings will be built jointly by Americans and Soviets, with the Soviets responsible for the structure and shell of the buildings -- to cost $54.6 million -- and the Americans completing the job. Today's ceremony comes more than a decade after the two countries in May 1969 agreed to swap sites in their capitals for new chanceries.

The Soviet site in Washington is at the old Mt. Alto property on Wisconsin Avenue north of Georgetown. The Soviets already have put up an apartment block.

The swap of the tracts was agreed to in 1972. But delays and differences that seem to reflect the underlying tensions of the bilateral relationship have plagued the U.S. project here for years. Indeed, final settlement of payment to the Soviets for their work was achieved only this summer when the United States balked at allowing the Soviets to move into their new apartments until agreement was reached on the cost here. The Soviets originally had demanded $80 million for their work.

The U.S. complex includes an eight-story chancery flanked on three sides by consular offices, 11 townhouses, 124 apartments, a Marine guard quarters, a school, warehouse and other facilities.

Since 1952, the U.S. mission has been housed in a former apartment building on Tchaikovsky Street, including about 40 apartments. Most Americans live in various guarded foreign compounds elsewhere in Moscow.

The new embassy is a few hundred feet from the present one. But its site, in a depression and on a side street, removes it from thousands of passers-by daily on the busy "Garden Ring" Boulevard, where the mission now is located.

"This is surely going to make our security easier," an American official said today, but he lamented what he called "an increase of compound fever" he suspects will occur after gathering so many employes to live and work in one place.

Over the years, the United States has disclosed major Soviet bugging operations, including the 1960 discovery of a microphone in the beak of the eagle on the official seal in an office; a 1964 disclosure of 40 hidden microphones found in the embassy; and last year's discovery of an antenna near the roof connected through a hidden shaft to a tunnel and electronic listening room beneath the building manned by a Soviet.

Meanwhile, the Soviets have bombarded the building almost continuously in recent years with microwave radiation. The State Department has said it has had no detectable effect on the staff, but narrow-mesh screens were put on all windows to block the waves.