combination wireless and ethernet home networking

I have a new dimension E510.  I have an older Dimension 4300S.  The new computer has a Dell Wireless 1450 USB adapter that I am not using becuase I am using Comcast cable high-speed internet for the new computer.  Comcast supplied their own (Netgear) gateway and wireless router.  I have the new computer hooked to the gateway via an ethernet cable.  I would like my older desktop (dimension 4300 S) to be able to be part of my wireless home network.  Can I install the wireless 1450 adapter on the 4300S and have it communicate with the Netgear wireless router?  (The 4300S doesn't have a PCI card slot, so I'm not sure I can use a regular wireless card without buying some additional hardware that gets me a PCI slot).  Since I already bought the 1450 wireless USB adapter but am not using it, I'm hoping that will work.  Just as an FYI, at one time I had an external DSL modem hooked up to an ethernet card (Linksys_) that I had installed on the 4300S when I was a Verizon wireless customer.  I'm assuming I can disable that and make the wireless connection work.  I would rather not be tied to an additional ethernet connection from the old desktop to the router.

 

Thanks for any advice and suggestions