Didn’t Russian papers made public after the fall of the USSR confirm this? I thought that was one of the revelations.
The Venona papers. These were intercepted embassy message traffic that was decoded and helped bust the spy ring. It was so classified it couldn't be used as a trial exhibit lest the Soviets catch on.
Among other things, the messages contained instructions to the the agents about managing the Rosenbergs and their co-traitors.
For decades, the Rosenberg's relatives and the left loudly proclaimed the innocence of Julius & Ethel, until the collapse of the USSR. The KGB archives and America's Venona papers both prove their guilt unambiguously.
The Venona project was probably instrumental in Joe McCarthy's exposing communists. Unfortunately, he ran roughshod over innocents - deliberately perhaps, lest the Soviets suspect if he exposed their spies unerringly.