CONTINENTAL FLT 458 (BOEING 737, N13331) WAS FLYING NE AT FL370 AS CONTINENTAL FLT 703 (BOEING 737, N432PE) WAS FLYING FM THE OPPOSITE DRCTN AT FL350. BOTH FLTS WERE ON THE SAME FREQ, BEING CTLD BY ATLANTA ARTCC (SECTOR 47) CTLR, WHO HAD JUST COME ON DUTY. IN PREPARATION FOR HANDOFF TO SECTOR 33, FLT 458 WAS TOLD TO DSCND TO FL 350; FURTHER DSCNT OF FLT 458 WAS INTENDED AS SOON AS IT WAS CLR OF A 3RD ACFT AT FL330. THE RELIEVED SECTOR 47 CTLR HAD AGREED WITH THE SECTOR 33 CTLR THAT SECTOR 33 WOULD GIVE FLT 458 A VECTOR SO IT COULD BE FURTHER DSCNDD. BASED ON AN ASSUMPTION OF WHAT WAS TO BE DONE, THE SECTOR 47 CTLR HANDED OFF FLT 458 TO SECTOR 33. DRG THIS TIME FRAME, THE SECTOR 33 CTLR WAS ALSO RELIEVED & FLT 458 WAS NOT VECTORED FOR ABT 3 MIN. SEEING A POTENTIAL CONFLICT, THE CTLRS TURNED THEIR RESPECTIVE FLTS, BUT BOTH WERE TURNED TOWARD THE NW (FLT 458 TO THE LEFT & FLT 703 TO THE RGT). AS BOTH ACFT WERE TURNING, THEY CONTD TO CONVERGE. SUBSEQUENTLY, THE CONFLICT ALERT SOUNDED & THE ACFT PASSED EACH OTHER AT THE SAME ALT WITH ABT .6 MI SEPN.