need advice for attaching "grill" to mirror frame

I am attempting to reproduce the following mirror frame:
http://wrightcatalog.stores.yahoo.net/ellismirror.html

Everything looks simple, but I do not know how I should attach the
interior grill to the outer mirror frame. I am going to make the
grill 3/8" thick and it will be flush with the outer surface of the
frame. The frame will be 3/4" thick with an inner 3/8" rabbet to
accept the mirror. That leaves a 3/8" lip on frame where the grill
will attach. There seems like there should be a customary way of
attaching this grill to the outer frame (other than gluing it to the
mirror itself). Any suggestions?

Thanks