A traffic conflict resulted when an A319 Captain misheard and acted unilaterally on a clearance to fly a 280 degree heading as a climb to FL280. The report included an effusive mea culpa and re-dedication to CRM SOP.

Narrative

We were climbing through about 9;000 FT MSL. We were assigned to climb and maintain 10;000 MSL. We were advised of an inbound aircraft above us and descending at our 2 o'clock. We were still IMC at that time. The First Officer advised ATC that we were IMC. I then thought I heard ATC clear us to FL280 so I allowed the aircraft to continue climbing and I set 28;000 FT on the altitude alerter.As we approached 10;000 MSL; we got a TCAS RA to 'Monitor vertical speed.' We became VMC at that moment and I could see the aircraft at about our 11 o'clock and above us. I disengaged the autopilot; stopped the climb manually; and leveled the aircraft at 10;000 MSL. I remember that we reached 10;300 MSL during the maneuver. At the same time; I heard the ATC repeat to us that our climb clearance was 10;000 FT. The actual clearance was to fly heading 280 and maintain 10;000 FT; but I had heard it incorrectly.By the time the First Officer had time to call attention to my mistake; I had stopped the climb and started back to 10;000 MSL. I had not changed our heading; either. We were heading 266 degrees and the assigned heading was 280 degrees. The First Officer confirmed to ATC that we were at 10;000 MSL as we leveled there. The smallest vertical separation I noted on the TCAS was 900 FT and perhaps 3;000 FT horizontally; but I am not sure of the horizontal measurement. ATC did not make further mention.My mistake; I heard the clearance to turn to heading 280 and thought that it was a climb clearance to FL280. I did not clear the new altitude with the First Officer before commanding the aircraft to climb further.I will remember this episode forever. It was a wake up call. As we are supposed to do; I will clear newly assigned altitude assignments with the non flying pilot before initiating or continuing a climb or descent. I will try to listen to clearances more critically even if they seem routine.