I recently had the third row seat installed in my v70 2000. The seat does not stay up, both the part that goes into the back of the second seat as well as the part that goes into that keep on coming out.

Haas any one elso had this problem?

Common. The plastic female part that goes into the seat back needs to be epoxied so that it can't come out. If there's then any latching problem with the other part, it can be adjusted and sometimes needs to be during a proper install job.

That "female piece" was designed to snap in place snugly into the seatback. Assuming this piece was never fully engaged, you might want to try shaving away some of the carpeting material, to gain a cleaner contact area for the piece to grab onto. I had a tough time getting mine to lock in, had to use some direct and firm force.

Before snapping it in place into the seatback, make sure it isn't upside down. That may affect how the longer piece from the 3rd seat catches into the receiver.
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-Fred, black '00 V70R

I know how it was designed BUT being a guy who's installed 3rd seats in Volvos for ages (before 850s), MANY times I've seen those fall out after that firm snap-in done by many others despite ones carpet cutting skills SO epoxy works everytime, it NEVER comes out again and that's also never a problem.