I have a web site running on IIS on my localhost. This web site has directory security set to only allow Integrated Windows Authentication. It is part of an intranet and nee
AS noted by Pontus Gagge, IIS needs to pass a Kerberos ticket to SQL Server. That was enough to tip my Google-fu in the right direction.
Firefox supports Kerberos, but, you have to tell it which domains it trusts to send the Kerberos tokens too.
Note: This is not the same as gbn's solution which just configures firefox to not prompt you to enter domain account details on login.
Also, if you have already tried to authenticate through the stack in your current Firefox session, you will need to restart Firefox for this to work.