A320 FLT CREW REPORTS PRESSURIZATION ANOMALY AT FL340 AND AFTER DESCENT ELECTS TO DIVERT TO ENROUTE ALTERNATE FOR MAINTENANCE.

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DURING CRUISE FLT AT FL340 IN NIGHT VFR CONDITIONS; WE GOT A REPETITIVE CHIME AND ECAM INDICATING THE CABIN ALT WAS CLBING (CAB PR-EXCESS CAB ALT). AS PER THE ECAM ACTION ITEMS; THE FO AND I DONNED THE OXYGEN MASKS AND ESTABLISHED COMS BTWN EACH OTHER. WE NOTICED THAT THE CABIN WAS INDEED CLBING; BUT SLOWLY. ON THIS ACFT; THE #1 CABIN PRESSURIZATION SYS FAULT HAD BEEN WRITTEN UP AND RESET TWICE PRIOR. AT THIS POINT; WE DECLARED AN EMER WITH CTR AND REQUESTED A DSCNT TO 10000 FT. I WAS THE PF AND MAINTAINED CTL OF THE ACFT WHILE THE FO HANDLED THE RADIO. AS WE RECEIVED AND BEGAN OUR DSCNT TO 10000 FT; WE CONTINUED THE ECAM ACTIONS UP TO DEPLOYING THE PAX OXYGEN MASKS. EVEN THOUGH THE CABIN WAS CLBING; IT NEVER GOT ABOVE 6450 FT AND NEITHER WE NOR THE FLT CREW EVER FELT ANY PRESSURE CHANGES. WE MADE THE DECISION NOT TO DEPLOY THE PAX OXYGEN. WE ALSO MADE THE DECISION TO DIVERT TO ZZZ APPROX 80 MI AWAY. DURING THE DSCNT; WE ADVISED THE FLT ATTENDANTS WHAT WAS HAPPENING; THAT WE WERE DIVERTING; AND TO PREPARE THE CABIN FOR A NORMAL LNDG. WE ALSO ADVISED THE PAX AND CONTACTED THE STATION IN ZZZ. AFTER DSNDING THROUGH 15000 FT; WE GOT THE CAB PR SYS #1 AMBER FAULT ECAM. THROUGH 14000 FT; THE EMER ECAM DISAPPEARED. AT 10000 FT; WE WENT OFF OXYGEN; ADVISED CTR THAT THE ACFT WAS FLYING NORMALLY AND WE'D NEED SOME TIME TO RUN THROUGH VARIOUS CHKLISTS BEFORE LNDG. AT THIS POINT WE HAD ONE OF THE COMPANY'S NON-REVING CAPTS COME UP TO THE COCKPIT FOR ANY ADDITIONAL HELP AND VERIFICATION OF PROCS THAT MAY ARISE. IN ADDITION TO THE QRH PROCS FOR THE DSCNT; WE HAD TO RUN THE OVERWT LNDG CHKLIST AS WE WERE LNDG OVERWT. ALL ACTIONS AT THIS POINT WERE ACCOMPLISHED IN A RELAXED; YET TIMELY; FASHION AS THE ACFT WAS FLYING NORMALLY AND WE HAD BETTER THAN 3 HRS OF FUEL ONBOARD. AFTER ALL THE QRH AND NORMAL PROCS AND CHKLISTS HAD BEEN ACCOMPLISHED AND VERIFIED; WE ADVISED ATC WE WERE READY FOR THE APCH. WX AT THE TIME WAS CLR WITH CALM WINDS. ATC HAD ALREADY ADVISED ZZZ FIRE RESCUE TO STAND BY. APCH AND SMOOTH LNDG (AS PER QRH) WAS ACCOMPLISHED WITHOUT INCIDENT. WE TAXIED TO THE GATE WHERE CUSTOMER SVC CREW MEMBERS AND CONTRACT MAINT PERSONNEL WERE STANDING BY. MAINT FOUND THE #1 CABIN PRESSURE COMPUTER TO BE FAULTY. IT WAS SWITCHED OUT WITH THE #2 COMPUTER AND THE SYS CHKED OPERABLE. THE OVERWT LNDG INSPECTION FOUND NO DAMAGE. THE LOGBOOK WAS SIGNED OFF PER CONTRACT AND ACR MAINT. THE FLT CONTINUED ON TO THE ORIGINAL DEST WITHOUT FURTHER INCIDENT.