A DC8 CARGO FLC HAD A NEAR MISS WITH AN AIRBUS A320 FLT NEAR MIA.

Narrative

I WAS THE PF AND WAS DODGING TWRING CUMULO NIMBUS CLOUDS REPLETE WITH LIGHTNING. AS THE CLRNC WAS ISSUED; ALL I RECALL WITH CERTAINTY IS THE 'DIRECT MIA; DSND....' BEING ENGROSSED WITH THE FLYING OF THE AIRPLANE AND AVOIDANCE OF TSTMS; I DO NOT RECALL THE ALT TO WHICH WE WERE CLRED. THE CAPT; WHO WAS HANDLING THE COMS; SET THE ALT IN THE ALT REMINDER AND I CONTINUED OUR DSCNT TO THAT ALT. CONCENTRATING ON THE IMMEDIATE PROBS OF FLT; THAT IS THE WX AND TURB; I MISSED THE CAPT'S CLRNC READBACK AND ASSUMED THE SETTING OF THE ALT REMINDER WAS CORRECT. THE CAPT WAS ALSO INTENSELY INVOLVED IN WATCHING THE WX RADAR AND OFFERING SUGGESTIONS AS TO THE RTE OF FLT. DURING THE DSCNT I RECALL HEARING SEVERAL COMS BTWN ATC AND AN ACR FLT WHOSE CALL SIGN I DO NOT RECALL. THE ESSENCE OF THE COMS WAS WHETHER OR NOT THE ACR HAD VISUAL CONTACT WITH AN OPPOSITE DIRECTION DC8. I RECALL THE ACR SAYING THAT HE WAS IFR. I CONTINUED THE DSCNT 10000 FT; THE ALT THAT WAS SET IN THE ALT REMINDER. MORE OR LESS; COINCIDENT WITH LEVELOFF; I RECALL ATC QUERYING US IF WE WERE LEVEL AT 11000 FT. I SEEM TO RECALL THE CAPT ACKNOWLEDGING IN THE AFFIRMATIVE; 'ROGER; LEVEL AT 10000 FT;' AND THE FE QUERYING THE CAPT ON 10000 FT VERSUS 11000 FT. THE CAPT ADVISED THAT WE WERE AT 10000 FT. ATC ADVISED THAT OUR ASSIGNED ALT WAS 11000 FT. IMMEDIATELY; THE CAPT ASKED IF ATC WANTED US AT 11000 FT AND I IMMEDIATELY INITIATED A CLB TO 11000 FT. ATC REPLIED THAT 10000 FT WAS OK AND I RECAPTURED 10000 FT. THERE WERE OTHER RADIO XMISSIONS BTWN THE ACR FLT AND ATC RELATING TO THE PROX OF OUR ACFT. AVOIDING THE PROXIMATE WX AND MAINTAINING WHAT I BELIEVED TO BE THE ASSIGNED ALT WAS FIRST AND FOREMOST ON MY MIND. BEING IN A 'RADAR ENVIRONMENT' AND IN IMC; I ALLOWED THAT TFC SEPARATION; TO A GREAT EXTENT; WOULD BE MANAGED BY ATC. IN RETROSPECT; WHY DIDN'T I CONFIRM THE DSCNT ALT? MY RECOMMENDATION: REQUIRE TCASII ON ALL ACFT.