Try giving the desktop PC a fixed IP address in the same range as the Livebox. For example, if the Livebox has the address 192.168.1.1 then give the PC addresses such as:
IP address - 192.168.1.5
Subnet Mask - 255.255.255.0
Default Gateway - 192.168.1.1 (the Router's IP address)
DNS Server - 192.168.1.1 (the Router's IP address)
(Assuming Windows XP)
To do this, open the Network Connections folder, right click the Wireless Network Connection and select Properties. Highlight the entry for "Internet Protocol(TCP/IP)" and click the Properties button.
In the next Properties sheet that opens, click "Use the following IP address" and type in the settings above. If your Livebox *DOESN'T* have the address 192.168.1.1 then post back with the actual address if you need further help.