A MCDONNELL DOUGLAS DC-10 (VIASA FLT 800) & AN AIRBUS A310 (PAN AM FLT 221) WERE INVOLVED IN A NEAR COLLISION WHILE ON INTL FLTS OVR THE ATLANTIC OCEAN ON (AIRWAY) AMBER 300. BOTH ACFT WERE OPERG UNDER IFR & WERE IN A NON-RADAR ENVIRONMENTW/O DRCT RADIO CTC WITH AN ATC FACILITY. COMMUNICATION BTN THE ACFT & ATC WAS THRU ARINC, A COMMERCIAL COMPANY. WHILE SOUTHBOUND AT FL 370, FLT 221 MADE A REQ THRU ARINC TO CLIMB TO FL 390. WHEN THE CONTROLLER (CTLR) AT NEW YORK CENTER OCEANIC CTL RCVD THE REQ, HE COORDINATED WITH SAN JUAN ARTCC & SCANNED HIS BOARD, BUT SAW NO CONFLICTS & APPROVED THE REQ. AFTER LVLG AT FL 390, FLT 221 CONVERGED HEAD-ON WITH FLT 800 WHICH WAS NORTHBOUND AT THE SAME ALT. THE AIRCREW OF FLT 800 TOOK EVASIVE ACTN; THE CREW OF FLT 221 ELECTED NOT TO MNVR. THE CAPT OF FLT 800 SAID HE ESTD THE ACFT PASSED WITH 500' TO 1000' HRZNTL SEPN & 350' VRT SEPN. THE CTLR HAD JUST COME ON DUTY. AFTER FLT 800 HAD PREVIOUSLY PASSED A MANDATORY RPRTG POINT, ITS STRIP HAD BEEN REMOVED FM THE CTLR'S BOARD SINCE THE NEXT INTXN WAS A NON-RPRTG FIX.