Setting a router to be an access point

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Ah, I see! Serves me right for only reading half of your post and assuming no-one would be crazy enough to try to set up a wireless router as a client.

With some luck, you should be able to find a "client mode" setting in the admin control panel of your secondary router. If you do, you will have to input the SSID, MAC address and/or channel of your Verizon router to establish a connection. I haven't encountered any wireless routers that support client mode out of the box so far, but with some luck, you'll be able to download hacked firmware for it. If it's a Linksys router, look into Tomato.

Also, if you do find a client mode setting, you should disable the WPA encryption on your Verizon router to allow your client router to connect to it.

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Edit: If you can't get your secondary router to work as a client and you don't want to buy another access point or router, you should consider getting a single, inexpensive wi-fi card for your computer and share its connection with your Xbox 360 via your Ethernet card. CNet sells a pretty decent card for $20. Unlike approximately all client mode-bestowed APs and routers, it will also let you retain your WPA encryption. If you want to use this approach and the computer is running Windows, look into Internet Connection Sharing.

The drawback with the aforementioned method is that it requires your computer to be on whenever you want to be able to connect to your network with your Xbox 360. It may save you quite a bit on your electric bill by investing in an excellent and stable wireless router that works well with client mode (with said Tomato firmware), such as the Linksys WRT54GL. However, it won't connect to encrypted wireless networks.