There isn't an old ring unless it's been replaced before, the housing starts out as a single molding that you separate by cutting through it, the ring then reattaches the part you cut out, but you can do it with screws and glue/sealant, like the video above shows.

 

If you do end up needing to replace the blower motor, & you're not an Amazon Prime member, Rock Auto has the "Four Seasons" brand blower motor at around $49 shipped, as well as other brand options for more $$$.