@cassielevett, you'll need to kerberise your SharePoint environment by allowing "claims" based authentication.

It may not be possible depending on your Sharepoint implementation. You may be stuck with NTLMv1 and keychains.

But yes, kerberos is the only option for "SSO" which is what you are describing. There is no secure way to fudge that or script your way around it.

We have Kerberos implemented in some degree. Sharepoint prompts for a password when going to our intranet. I don't know all the details on that side but I can find out.

Is there anything I need to do on the browser / client side?

NTLMv1 Sharepoint is a complete nightmare.

It should "just work" if it's implemented correctly on the site. Whenever you log into the system you should get a Kerberos ticket and then Sharepoint would never prompt for a password. Proper Kerberos never needs your password outside of the system's login/screensaver prompt unless you work for like 10 hours straight and need the ticket refreshed.