An air carrier Captain flying with a non-operating auto pilot reported an altitude deviation when the second autopilot failed.

Narrative

The airplane had an MEL for a malfunctioning autopilot B. We were using autopilot A. I was pilot flying and we were descending via the ZZZZZ arrival into ZZZ. Descending through about 26000 ft; we both noticed that the autopilot was starting to descend a little bit faster than expected. However; we thought it was just being a little aggressive to make the bottom of the FL220 restriction at ZZZZZ1. However; shortly afterward; we got a STAB OUT OF TRIM light on the panel and the autopilot continued to descend even more rapidly and speeding up well below the path in VNAV PATH. I pulled the speedbrakes to slow the descent as we started to go through 320 knots. The FO (First Officer) pulled up the QRH and the next step was to disconnect the autopilot and hand fly. Before disconnecting the autopilot (I knew it would be out of trim and perhaps cause some abrupt pitch changes); I turned on the fasten seatbelt sign; put on my shoulder harness; and told the flight attendants to take their jumpseats.I then disconnected the autopilot which was way out of trim and used the electric trim to correct the nose down pitch attitude. We went below the 17;000 ft. restriction at ZZZZZ2 and I had the FO tell ATC we needed to level off due to an issue. I leveled off at 16500 ft. and got the airplane stabilized. I then told the FO to request a descent and vectors due to an autopilot issue instead of trying to meet the VNAV requirements of the arrival. After he got the vectors and a descend and maintain clearance; I handed the controls to him and told him to keep flying while I went over the checklist. It was a short checklist; but I wanted to verify everything had been accomplished and I briefed the flight attendants. I completed the checklist which only said not to re-engage the autopilot since the electric trim functioned normally. I gave the flight attendants a briefing since it was a stab trim issue but I told them to expect a normal landing. I also confirmed there was no injuries during the autopilot disconnect since it was very out of trim when I disconnected the autopilot and jerked the plane upon the disconnect. After all of that was accomplished; I took the plane back and we executed a visual approach to XXR and landed safely. I then wrote up the issue and made a verbal call to maintenance upon arrival to the gate.