MacBook won’t connect to Internet through wireless router

We have a Netgear WGT624v3 Wireless Router connected to our cable modem. We have one MacBook that can connect to the local network (Airport shows a connection with strong signal strength to our network) but cannot connect to the Internet. Safari gives a “You are not connected to the Internet” error when trying to surf the web. Apple Care has been no help at all. Their suggestion was to buy an Apple Router. MacBooks Network Diagnostic Tool always says that the network connection is working fine. MacBook can connect to Internet when plugged into the router. Any suggestions?