Eastern Airlines Flight 92, a Boeing 727, N8840E, and an Air Force KC-135, Reg #60-0330, call sign FOWL 24, were involved in a near collision while both aircraft were maintaining FL350. The captain of Flight 92 reported that during the occurrence, the KC-135 passed directly in front of his aircraft at a distance of about 1/4 mile, crossing from right to left. Also, he reported that there was insufficient time for him to avoid wake turbulence from the KC-135. The KC-135 flight crew were unaware of the occurrence until after they had completed their training flight. An investigation revealed that the incident occurred after Jacksonville Center had instructed the KC-135 crew to contact Atlanta Center during a hand-off. The KC-135 copilot thought he had complied, but no calls were recorded until after the incident. The Atlanta controller was not aware of the KC-135 in his assigned airspace until after the captain of Flight 92 reported the near collision. The investigation revealed that radar data on the KC-135 was available and displayed on the controller's scope.