You could try giving the Desktop PC a fixed IP address in the same Subnet as the Router (Home Hub). I believe the Home Hub has an IP address of either 192.168.1.1 or 192.168.1.254 so, assuming you're running Windows XP:
Open the Network Connections folder ("Start > Run", type NCPA.CPL and click OK), right click the Local Area Connection and select Properties. Highlight the entry for "Internet Protocol (TCP/IP)" and click the Properties button.
In the box that opens, select "Use the following IP address" and type in:
IP address - 192.168.1.5
Subnet Mask - 255.255.255.0
Default Gateway - 192.168.1.254 (the *ROUTER'S* IP address)
DNS Server - 192.168.1.254 (the *ROUTER'S* IP address)
If you are unsure what the Router's IP address is then, in the computer that does connect, open a Command Prompt window ("Start > Run", type CMD and click OK) then, at the prompt, type...
...and press Enter. This will display details of all network adapters in the computer. Look for the "Default Gateway" address, which will be the Router's IP address.
To revert to automatic addressing, set BOTH the IP and DNS sections to get their addresses automatically.