Xbox powers on, no signal, shuts off after a minute

Sorry to hear it. This does sound as though it's a hardware issue and would need to be repaired or replaced. You can set up a repair of the console at https://devicesupport.microsoft.com.
Once it's set up and you get your shipping label via email you can pack up the console. Please be sure to only send the console and nothing else as Xbox does not guarantee that extra items such as cables or controllers will be returned.

However, if your console is the original Xbox One with the external power brick then I'm afraid that they are no longer serviced by Xbox

Hi DarthMattster

Thank you for posting on the Xbox forums. I'd be happy to help!

It's very likely that the power supply has given out if the console turns off after a minute. The console's OS could force a shutdown to prevent heating, but the console would not be in the case if it only stays on for a minute.

In the diagnostic screen, I would recommend doing a complete factory reset removing everything from the console if possible, however if the console powers off in the process or the reset cannot complete, then you will need to set up a repair order online
to ship out your console to get repaired. https://support.xbox.com/my-account/warranty-and-service/getting-your-product-serviced

There is a work around to this is you set the projector to turn off automatically after x minutes when there is no signal.

The only other option available is to have the xbox set to power saving instead of instant on. The ability to send the Power Off signal in this way is done through the Xbox One in Settings > Power & Startup > TV & A/V power options

Is your console in instant on or power saver mode. Switch it to the one it's not set to and hold the power button on the front of the console until it shuts off. After it shuts off unplug the power brick from the console and the outlet for a minute press
the power button the console 3 times after a minute plug everything back in and power back on and see if the issue is resolved.

Do a power cycle turn off the xbox holding the button for 30 seconds then after it shut off wait a few minutes then turn it on, it should stop the game from freezing