Mechanically it should be possible.  However, some caveats:

• The One is a 5V device, and has a different power supply.  You'll need to replace the power connector on the back.

• The accelerometer is mounted on the mainboard, which on the One is flat, but on the Classic is vertical.  You'd need to modify the accelerometer daemon to take this into account.

• There is no chumbilical populated on the One, but the pads are there.  This means the front bezel might not stay in place as easily, but it's nothing glue wouldn't fix.

• The wifi riser is different.  On the One it's a card, and on the Classic it's a ribbon cable or a USB cable.  An easy solution is to solder on another USB port to the pad on the mainboard and slide the wifi card in there.  If you have a 3.7 Classic, you may be able to slice off the end of the USB cable and stuff the wires into the One's USB connector socket.

• The chumbilical pinout is different.  On the One there are pads for a connector, but the pinout is subtly different.  You'll want to bring over only the USB and headphone pins, not the other pins.

• I'm not sure what you'd do with the speaker.  Probably just send the one speaker to the two outputs on the back.

So while it's easy to mount, getting all the connectors to line up is the biggest challenge, owing mostly to the lack of a populated chumbilical connector.  If you can work around that, you're good to go.