BE36 PLT TOOK EVASIVE ACTION TO AVOID A COLLISION WITH INTRUDING ACFT IN ZMA CLASS E.

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FLYING IFR BOCA RATON (BCT) TO FALCON FIELD (FFC); IN DENSE CLOUD IMC; MY SKYWATCH (TCAS) SHOWED RAPIDLY CLOSING TARGET ON INTERCEPT AT 10 OCLOCK; TARGET FIRST SCOPED AT 3 MILES; 300 FT BELOW. ATTEMPTED CALL TO CTR STEPPED ON. TARGET CLOSED TO 1.5 MILES; 200 FT BELOW WHEN CTR REACHED. I REQUESTED INSTRUCTION REGARDING INTERCEPTING ACFT. CTR DID NOT APPEAR TO UNDERSTAND MY CONCERN. I EXPLAINED THAT TARGET WAS LESS THAN A MILE AND APPARENTLY CLIMBING; ON DIRECT INTERCEPT. AFTER SEVERAL SECONDS WAITING ATC RESPONSE; I INFORMED CTR I WAS INITIATING URGENT 500 FT CLIMB TO THE R SHORTLY AFTERWARD; TARGET PASSED BELOW ME; SHOWING 100 FT BENEATH ON SKYWATCH. CTR INFORMED ME THAT SEPARATION WAS A FULL 500 FT. I REMAINED IN ZERO VISIBILITY (NEVER SAW OTHER ACFT) FOR 3 MORE MINS. RETURNED TO ASSIGNED COURSE WITHIN 5 MINS. I WAS AUDIBLY ALERTED BY TCAS; AND IN ABSENCE OF CLARIFICATION BY CTR; TOOK WHAT I FELT WAS APPROPRIATE AVOIDANCE ACTION IN ZERO VISIBILITY. CTR WAS QUITE BUSY AT THAT TIME; AND CLOUD COVER (IMC) WAS SCATTERED AND INFREQUENT.