You need to pull up the center console cover first. Working from front to back carefully pull up to disengage the six pressure clips. You may need to disconnect the traction control wiring connection if you have one. The cup holder is attached to this panel and you can see how to repair or replace it.
On my 1999 Regal the cup holder swing out side piece with the little spring fell off. I was able to retrieve them from under the console. Took me a few minutes to figure out how to put it back and re-engage the little spring. I added tape to secure cup holder top piece to the base to prevent them from separating again. Reattach the traction control wire clip and carefully push down to resecure the console panel.
I believe the same Regal console panel fit from 1997 to 1005 model - but please verify that. Good Luck!
Additional comments: I had removed and fixed the cup holder just as described about, but after a year or so the tape failed and it was broken again. Inspecting the cup assembly closely, I saw the factory joint between the top and the bottom piece of the cup holder were melting a few pieces of plastic tabs. When the plastic tabs breaks, the side hinge assembly and the springs falls off. The top pieces and the bottom pieces butted together. Perfect to add some super glue to glue them together instead of using tape. Since you have the unit out, may as well check the front joint where it attached to the door. It is also another melted plastic tabs. I added super glue to that joint too and the cup holder has been trouble free for 3+ years now.
How do you replace the morror assembly for a 2001 Buick Regal
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The 2001 Buick Regal has 12 valves.
The 2001 Buick Regal has a V6 engine.
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The 2001 Buick Regal has overhead valves (OHV).
The 2001 Buick Regal is a gas-powered vehicle.
The 2001 Buick Regal has a 4-speed automatic.
The height of the 2001 Buick Regal is 4 ft. 8.6 in. (56.6 in.).
The 2001 Buick Regal runs on regular unleaded.
The 2001 Buick Regal is 16 ft. 4.2 in. (196.2 in.) long.