problem with Mail & Gmail passwords

I am having problems with Gmail passwords. I have two gmail accounts and never had any problems. But recently — just when I bought a new iMac, I don't know whether it is a coincidence or not (I used to have a MacBook using Tiger 10.4.9) — I cannot seem to get the computer to store both passwords: I am constantly being asked for one password or the other. It is a real pain. I have spent hours configuring and reconfiguring (getting rid of an account in Mail and setting up a new one, changing the password at the gmail website, etc, but nothing has helped).

I would really appreciate it if someone could enlighten me on this. Has something in Gmail changed recently, or is it a problem with Leopard or a problem with the recent version of Mail?