Dead battery doesn't matter if the power adapter is in. You can take the battery out completely and it runs fine.

First thing to check: plug the adapter into another PB, which you know to be working properly, and see if it shows power coming in. (If it's running OS X, look at the icon near the right end of the menu bar. The D-cell-shaped icon will show an X for battery bay empty, solid bar indicating percent of charge for running on the battery, and either a sine wave or a wall plug for external power coming in.)

If the power adapter checks out it could be many things, starting with the connections where it plugs in. At that point I'd go to a service person. (You might try pressing Reset (tiny round button just inboard of the modem port under the port cover at the rear, use a paper clip to push it), but it probably won't do anything.)