GA pilot reported an engine failure in flight when one fuel tank ran out of fuel and they were unable to select the other tank. Pilot eventually was able to move fuel selector and restart engine during descent.

Narrative

I was conducting a cross country trip from ZZZ1 to ZZZ. This was on the final leg of the multi leg trip. I normally alternate between the left and right tanks every 30 minutes; however on this leg I followed the Piper manual's guidance for maximizing endurance; which instructed the pilot to run one tank dry; then switch the fuel selector to the other tank. This would ensure that all of the usable fuel in one tank was consumed and result in the maximum range. Additionally it would allow me to more exactly determine my precise fuel burn that day and determine if I needed to stop short of my destination at my predetermined bingo fuel point or if I could continue to my destination and land with my minimums of 1 hour fuel remaining.Around 2.6 hours into the flight; as expected the engine coughed and I went to select the other tank. However the fuel selector would not budge and the fuel selector handle was ice cold. I reached with both hands to attempt to turn the fuel selector but was unsuccessful; a few seconds later the engine was silent and the EGTs dropped. I immediately turned towards the nearest airport; calculated that I had enough altitude to glide to the airport and informed ATC of the issue. I remembered that during my preflight I had water in one of my fuel tanks when I sumped; and it had taken multiple attempts to obtain a fuel sample without water. I suspected that there was still a small amount of water in that tank; and it had settled to the bottom near the fuel selector and frozen; causing the stuck fuel selector. I informed atc what I believed the issue was and that I would continue to troubleshoot while gliding to the airport.Knowing the likely cause of the stoppage and understanding that the solution would be to turn the stuck selector; and believing that it was caused by ice and the cold temperatures at altitude I continued my glide towards the airport while I rubbed the fuel selector with my hand in an attempt to warm it up. Around 6;000 feet I was able to begin to turn the fuel selector and was able to select the tank with fuel remaining. In a few seconds the engine roared back to life. Satisfied that the issue was fixed; and I had a full tank that was now feeding the engine I changed my diversion airport to the second nearest airport that had services as the initial airport I selected did not have any services and continued the flight uneventfully. Upon landing I consulted with an Inspection Authority (IA); and we both agreed with my assessment of water freezing in the fuel selector and the ice melting as I descended and the air warmed.