A319 Captain reported a rapid descent after cabin pressure became uncontrollable. The flight crew requested priority handling and made a precautionary landing at destination airport.

Narrative

Trigger...at FL380 was given descent to FL340. At approximately FL360; the CAB PRESS status page displayed showing a cabin altitude of 9;500 feet and rising. Action requested priority handling with ZZZ ATC; started rapid descent to 10;000 feet; seat belt sign on. First Officer (FO) and pilot flying; performed a maneuver with the autopilot on. ATC gave us initial descent to FL270 and a frequency change. New Controller gave us descent to 10;000 feet with altimeter setting of 29.80. We were in VFR conditions; minimum safe altitude from charts approximately 5;000 feet. FO was able to descend quickly enough that a CABIN ALTITUDE ECAM did not trigger nor did the oxygen masks deploy in the cabin. I was able to speak with FA1; during the descent. FA1 confirmed that passengers were okay and did not report any injuries but would update us if something changed. I made a PA to passengers about aircraft condition and plan of action. Requested all to remain seated until in ZZZ.After level off and confirming aircraft was safe to continue; removed the aircraft alert from ZZZ ATC; but requested priority handling into ZZZ. Sent ACARS message to Dispatch. Stated our problem; action; current situation and plan. Received acknowledgment and new fuel remaining calculation for Denver at 10;000 feet. Highest cabin altitude was 11;350 feet. No ECAMs showed on the upper display. Outflow valve; inlet and extract vents all showed closed. During level off; safety valve opened when cabin diff pressure at -.5 PSI. Remaining flight was uneventful to a landing at ZZZ; XXR. Taxi to Gate at XX. Met by ZZZ maintenance and ZZZ FO representative.