Also, ensure that the card (s) in both machines are set up to allow connection to either
1)"Any available network (access point preferred)" or
2) "computer to computer (ad-hoc)".
These settings are under the advanced option on the Network Settings tab of Wireless networks.
The default can be "access point", which requires a router.
Re-reading your post suggests that the Internet PC is OK, it is just the new machine that needs it configuration checked.
Laptops, that haven't been locked down
, tend to allow connection to both ad-hoc and infrastructure networks.