Rated 5 out of 5 stars

Divrsit and Xander's fixes worked great for my non-hosted Gmail accounts, but then broke my hosted accounts. Below are all the steps combining both of their fixes, plus my fix, to make it work for everything.

1. Browse to C:\Users\YOUR USER NAME\AppData\Roaming\Mozilla\Firefox\Profiles\A RANDOM CODE IS HERE.default\extensions\
2. Locate the file {582195F5-92E7-40a0-A127-DB71295901D7}.xpi, rename it to {582195F5-92E7-40a0-A127-DB71295901D7}.zip and extract the files to a temporary location.
3. Open the components folder, look for the file gmServiceGmail.js and use a text editor to modify these 2 lines:
this._loginURL = "https://www.google.com/accounts/ServiceLoginAuth";
should be changed to this
this._loginURL = "https://accounts.google.com/accounts/ServiceLoginAuth?service=mail";

this._loginURL = "https://www.google.com/a/" + this.domain + "/LoginAction2";
should be changed to this
this._loginURL = "https://mail.google.com/a/" + this.domain + "/LoginAction2?service=mail";
4. Save the file, overwrite the existing gmServiceGmail.js
5. Open the file install.rdf with a text editor and change the line:
<em:maxVersion>4.0.*</em:maxVersion>
to
<em:maxVersion>6.0.*</em:maxVersion>
6. Save the file, overwrite the existing install.rdf.
7. Rezip all files you extracted in step 2 and name it {582195F5-92E7-40a0-A127-DB71295901D7}.xpi
8. Replace the old {582195F5-92E7-40a0-A127-DB71295901D7}.xpi for the 1 you just adjusted and restart FireFox