B757-200 CREW AND A B757 HAD LESS THAN LEGAL SEPARATION IN ATLANTIC INTL AIRSPACE.

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TCASII RA. WE WERE CRUISING SBOUND ON AMBER 300; LEVEL AT FL390. ABOUT 2 MINS S OF WAYDE; AT XA56Z; WE GOT A 'TFC; TFC' TCASII AURAL WARNING; FOLLOWED ALMOST IMMEDIATELY BY A 'CLB NOW' AURAL AND THE VSI COMMANDING A 2100-2300 FPM CLB. I IMMEDIATELY DISCONNECTED THE AUTOPLT AND STARTED CLBING; REFING THE VSI. REACHING FL397; A CLR OF CONFLICT AURAL WAS HEARD AND WE RETURNED TO FL390. WE THEN CALLED ZNY VIA THE BDA REMOTE (128.5) AND ZZZ COMRDO ON 113.96 TO NOTIFY THEM. THE FIRST TCASII CAUTION SHOWED AN ACFT TO OUR L; CLBING RAPIDLY ACROSS OUR FLT PATH L TO R. THE ACFT APPEARED TO BE A B757. WE TOLD ATC THAT IT LOOKED LIKE THE OTHER ACFT WAS MOVING E TO W; BUT CONSIDERING THE WINDS AT ALT; IT IS PROBABLE THAT THE ACFT WAS GOING N ON A300. SUPPLEMENTAL INFO FROM ACN 538370: PASSING WAYDE INTXN IN THE WATRS REGION; WE RECEIVED A TCASII TA. I BELIEVE THE TCASII DISPLAYED THE TFC AROUND 10 MI AHEAD; 1100 FT BELOW AND CLBING. I WENT EYES OUTSIDE TO GET A VISUAL AND PICKED UP THE ACFT AT 11 O'CLOCK POS. THE CAPT SAID 'HE IS STILL CLBING.' THE VISUAL IMPRESSION WAS ALSO OF AN AGGRESSIVE CLB. I LOOKED AT THE TCASII DISPLAY AGAIN AND THE TFC WAS 500 FT BELOW US AND CLBING AT AROUND 5 MI. AS THE CAPT DISCONNECTED THE AUTOPLT; WE RECEIVED A TCASII 'CLB' COMMAND OF 2000 FPM. WE CLBED TO FL395 BEFORE WE RECEIVED A 'MONITOR VERT SPD' WITH TCASII COMMANDING LEVEL FLT AND THEN VERY QUICKLY 'CONFLICT CLR.' WE PASSED 500 FT ABOVE AND 1-2 MI FROM THE TFC. I BELIEVE IT WAS A B757. I SAW NO AIRLINE MARKINGS. WE RETURNED TO FL390 AND INFORMED BERMUDA AND NEW YORK ZZZ COMRDO OF THE INCIDENT.