A320 Pilot reported loss of cabin pressure and a return to their departure airport where they made an overweight landing.

Narrative

Leveling off at 32000 ft.; we noticed cabin altitude on lower ECAM pulsating ADV message and indicating 9000 ft. Cabin altitude. Cabin vertical speed was continuing to climb at 100ft./minute while in level flight. Requested and received clearance to descend to 30000 ft. As power came off to descend; Cabin vertical speed increased to 500 ft./minute. Added thrust in descent to abate the increased cabin vertical speed; which then reduced to 100ft./minute. Cabin reached 10000 ft. while leveling off at 30000 ft. triggering ECAM Warning. Crew donned O2 masks; established communication and began priority descent to 10000 ft. Initially received clearance to 18000 ft. from ATC but prior to reaching; [requested priority handling] and received further clearance to 10000ft. Requested vectors and clearance back to ZZZ1 at 10000 ft. Finished QRH procedure; conferred with dispatch via SAT1 and informed Flight Attendants/ Customers of our pressurization issue and plan to return to ZZZ1. Issued Flight Attendants [instructions and threat status]. Prepared for overweight landing at 156000 lbs. on Runway XXL with Crash Fire Rescue (CFR) waiting in case of hot brakes. Touched down at less than 200ft./minute; applied max thrust reversers and minimal brakes. Exited XXL at Taxiway X; where CFR was positioned. Brake temp never climbed above 150 degrees. Taxied back to gate and handed aircraft off to Maintenance for repair. Customers were understanding as they deplaned. Assigned new tail; departed approximately 90 minutes later from [new] gate.