Can a router on one computer be used to connect to another router's Internet Connection?

I have a 3 room wireless network.

Room 1: Wireless B-Router (Desktop)
Room 2: Wireless G USB Adapter (Desktop)
Room 3: Built-in Wireless G Adapter (Laptop)

Room X: I have a Wireless-G Router in between all of them, it is connected to the internet (via my Cable Modem) but both of these devices are simply connected to them selves, the power outlet and the coaxial TV cable.

Room 1: ?????? Doesn't work yet (via Wireless-B Router
Room 2: Has internet access and home Wireless LAN access via the Wireless-G Routerin Room X (via USB wireless-G adapter)
Room 3: Has internet access and home Wireless LAN access via the Wireless-G Routerin Room X (via built-in wireless-G adapter)

Can I use a wireless-B router in Room 1 to connect to the Wireless-G router and use it's internet connection.  
To rephrase it, can the wireless-B router be used to connect to the wireless network, just like the USB Wireless-G adapter does?  I know that the wireless-B router will bottle neck at 11Mbps vs the wireless-G 54Mbps bandwidth.  But that's okay, I would rather use that then spend more money on another adapter.