First a rabbet was planed for receiving the glass (which is not real glass by the way), and next a small beading was planed on the front side.
The door frame was assembled with short tenons (1.25" long), and the beading was made continuous by mitering the corners of the beading /rabbet portion of the parts.
I removed the glass form the old door and adjusted its size so it would fit the new door.
The glass was mounted by means of some Sikaflex and 12 small screws that were driven in at an angle, so the head would press the glass down into the rabbet. Not exactly high end furniture style, but OK for this application.
A bit of the Siaflex oozed out, but given that the glass is plastic, I don't want to risk destroying it by being too eager with a chisel when scraping it off.
The two door latches were then mounted by chiseling out a recess, so I could attach the nut from the backside. They are not designed to be used on stuff much thicker than 1/4", so I had to remove some material from the front and the back.
Finally the door was mounted using the reclaimed hinges, and the door latches were adjusted so they could keep the door closed.
I have applied one coat of varnish to the door and the face frame, and as soon as it has dried, I plan to give it another coat, I think that will be sufficient, though the common rule of thumb according to boat varnishing in Denmark states that you should apply 7 layers of varnish with a light sanding in between. This is probably a rule of thumb that has been invented by the manufacturers of varnish!
New door, with first coat of varnish.
New door prior to varnishing.
Beading detail (and ugly mark from the clamp)
Face frame mounted.
Latch for the door.