Many DSL connections aren't "always on," but still have to dial up, in a sense (as compared to cable modems). It sounds like that /could/ be the case for you, because you said you were able to get it to work by just trying to connect repeatedly...
Assuming that's true, it sounds like you'd have something set to "dial on demand," which would try to initiate the connection once you tried to access the internet, which is consistent with my understanding of your description. Once it connects, I would imagine you would stay connected for at least two hours, assuming you didn't idle (many things could be set to disconnect if idle for X minutes).
Also, can you try this without the router, just to remove one variability, and see if you can get it to work directly connected to your computer?
As a side note: Pssst, Mac OS 9 is old. Upgrade to OSX

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