A320 Captain reported the cabin was not pressurizing properly and the flight crew opted to divert. The aircraft had a pack that was on MEL; leaving only one pack available for use on the flight.

Narrative

Upon arrival at the aircraft 65 minutes before departure time; we noticed that Pack 1 had just been placed on MEL due to recurring Pack 1 Regulator faults. While allowed per MEL; I do not like to fly over water on one pack; especially at night; and even more so when it's the first flight on a single pack. So I contacted Operations Control via Dispatch to ask for a different aircraft. Operations Control said none were available. I also emailed the Duty Chief Pilot for assistance but they never got back to me before departure time. I considered refusing the aircraft; but the FO (First Officer) and I chose not to. We planned to monitor the pressurization system more closely; knowing it was degraded. We boarded and departed on time. Initial climb-out was uneventful. Passing 10;000 ft.; we took a close look at the pressurization parameters and all looked normal. We continued to climb to FL290. We were busy in the climb conducting the numerous procedures required for Class II operations. After a few minutes at FL290 we were cleared to climb to FL330.As we initiated the climb; the Lower ECAM switched to the PRESS page and displayed the Cabin Altitude Advisory; flashing at 8;800 ft. and climbing. I pushed the V/S knob to level off at approximately FL294. It appeared that with Pack 1 OFF per MEL; our one remaining pack was 'weak' and not able to fully pressurize the aircraft. We noted the outflow valve was closed. No ECAM cautions or warnings were present.Since the cabin altitude continued to climb even while level at FL294; I directed the FO to request an immediate descent to 10;000 ft. ZZZ Center cleared us right away to descend and maintain 10;000 ft. I selected 10;000 ft. in the FCU (Fuel Control Unit) and began an open descent. At this point the cabin altitude indicated above 9;550 ft. and turned red; so I proactively announced emergency descent" even though the cabin altitude was still below 10;000 ft. The FO read the QRC checklist and I began a descent with autopilot off and spoilers fully extended. At the same time I stated out loud; 'We need to go back to ZZZ.' The FO advised ATC of our descent and the desire to return to ZZZ. We were additionally cleared to turn right direct ZZZ. Once the descent was established; we began the right turn towards ZZZ. I noticed no traffic at all within 20 NM of the aircraft during the descent and turn back via TCAS on ND (Navigation Display); and out the windows. I selected Terrain on ND to ensure we were over water as expected. It was a pitch black night with no horizon or visual cues.As we descended; the cabin altitude continued to climb above 9;550 ft. but at a slower rate. Then approximately halfway through the descent; the Master Warning sounded as the cabin altitude reached 10;000 ft. Shortly thereafter the cabin began to descend as the aircraft got lower. I recall the cabin was only 100 - 200 ft. above 10;000 ft. for approximately 20 seconds. We did not don oxygen masks when the cabin altitude reached 10;000 ft. due to the high workload of hand-flying the descent on instruments and the transient nature of the excessive cabin altitude. We felt the priority was on maintaining safe control of the aircraft. A320 FCOM (Flight Crew Operations Manual) 1 paragraph 5.1.2 states; "the pilot flying will be tasked with verifying the non-normal condition and confirming procedures… (but) it should not distract the pilot flying from his primary duty of safely flying the aircraft." However upon reflection; if the situation was progressing to the point where I thought declaring "emergency descent" was warranted; putting on our oxygen masks before beginning the descent would have been the better course of action. Upon leveling at 10;000 ft. we slowed to 250 kt.; re-engaged the autopilot; verified we were navigating direct to ZZZ as cleared; and assessed the situation. The cabin altitude was below 10;000 ft. and cabin vertical speed was at -250 FPM; descending. In fact as we transited back at 10;000 ft.; the cabin continued to descend normally and reached field elevation before we landed. We heard no unusual sounds indicating a leak of any kind. I next called the cabin to advise them of the situation and to check in on the cabin condition. They said everything was okay and the oxygen masks did not deploy. I told the F1 we were returning to ZZZ; landing in about 25 minutes. I said the landing would be normal but CFR (Crash Fire Rescue) may meet the aircraft. Next; we advised Dispatch via ACARS of our descent and plan to return to ZZZ. We initially set up the FMS for a return to ZZZ and ran the Overweight Landing Checklist. Planned landing weight was 154;000 lb. I then asked Dispatch if they preferred we go to ZZZ or ZZZ1; in case one was better than the other for our situation. Dispatch told us Operations Control preferred we go to ZZZ1 to facilitate an easier tail swap. We coordinated with ZZZ Center to divert to ZZZ1 and we reprogrammed the FMS again for ZZZ1. I continued to fly the aircraft and talk with ATC while the FO reprogrammed the FMS both times. ATC [provided priority handling] for us due to the descent; diversion; and planned overweight landing; and we concurred. I advised the flight attendants of our new plan to go to ZZZ1 and asked them to make a PA since we were in a descent below 10;000 ft. at this point and our workload was very high. The rest of the descent; approach and landing; ILS XXR ZZZ; were normal. Medium brakes were used per [the] software and max reverse per the Overweight Landing Checklist. Touchdown was in the touchdown zone at 200 FPM or less. ZZZ emergency equipment was dispatched to a location near the runway but they never approached the aircraft. We exited [Runway] XXR at [Taxiway] XX and taxied to [Taxiway] XY Gate XX normally.Brake fan use was delayed until brake temperatures reached 400°C on the left main gear; approximately 4 minutes after landing. Winds were 320 at 16 kt. for landing on [Runway] XXR. Brake temperatures never went much above 400°C and then began to cool. At the gate we wrote up the loss of pressurization and overweight landing. We debriefed Maintenance Control crew members face-to-face. We never ran the 'Diversion' checklist since both ZZZ and ZZZ1 are primary and well-known cities - we prioritized other necessary tasks. I provided the flight attendants all the information required for an event but I believe I may have failed to speak the words 'event' directly. The flight attendants confirmed with me after the flight that it was clear to them that it would be an event/normal landing."