Not quite sure from the description, but it might be a duplicate IP address issue. Try powering off your Sonos CONNECT, and while it is powered off, reboot your router. Wait a couple of minutes for the router to come back up before plugging back in the Sonos CONNECT, and a couple of more minutes before testing.  If you’re just powering off the Sonos, and not rebooting the router, the router is holding on to that potentially skunked IP address, so when you plug back in the Sonos without rebooting the router, you’re just getting the same IP as before. 

Hey Bruce. Thank you for the advice. It did not work for me. Again, the music played for one minute, then the Connect dropped out and disappeared from the system listing on the iPhone controller as well as disappeared from the “Manage System” listing. A few minutes later, the Connect reappears in both places. 

Could be this bug if the usual troubleshooting doesn’t work and you think the symptoms match. Effects connect amps but also reported by connect owners. 

Yes. I have read about several people having this problem. 
I will try posting about it to Sonos on the “X” platform. Sometimes, posts get an answer. 

So, I unplugged the Connect from power. Unplugged my router. When router finished rebooting, I plugged the power back into my Connect. The Connect still dropped off the controller. Next, I did a factory reset on the Connect. When I opened the Sonos app on my iPhone, it looked like the my existing system was gone. But…the app asked if I wanted to add the Connect. I followed through and now have my Connect functioning and playing music out of my Klipschorn speakers without dropping off the controller. All of my music services were gone also. Only problem is the now Sonos cannot connect to any of the Sonos speakers I own. 
When I use the new Sonos app on my laptop or old iPhone, my “old” system is still there untouched allowing me to play music on all my Sonos speakers. 
I have no idea how to resolve this. For now, I am not doing anything. 
I have spent more time and experienced more frustration than is warranted for expensive audio equipment.