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Yes you should be able to add the WG602 to your existing router to enable wireless. You'd then have to log into that access point from your computer via the wired network, usually using your internet browser (internet explorer / firefox etc) to set up the connection code and password. If you don't set that up, any tom dick and harriet will be piggy backing off your network.
But you might find the wireless access point costs a similar amount to your getting a replacement, integrated wireless router. That will allow you to still plug your PC in wired, but the same box will also do all the wireless stuff and the one setup process would do all the password stuff too.