<p>We want to switch a web server from Windows 2003 to Windows 2003 Enterprise (64 bits) to use 8GB of RAM. Will IIS 6.0 and an ASPNET 1.1 application be able to benefit from the change?</p>

My understanding is that there was a virtual address space limitation of 3 GB in ASP.NET 1.1, and that it was never made 64 bit compatible, though 2.0 was.
You can get IIS 6.0 to run 32 bit (i.e. ASP.NET 1.1) on the 64 OS, but it will be in a 32 bit mode (along with anything else hosted, including ASP.NET 2.0 sites).
Microsoft article on switching between 32 bit and 64 bit