annmemery I am not sure which iPod you really have. It sounds like a iPod Classic but you referred to it as a "1st gen iPod" which would be a totally different beast.
From your description it sounds like you will have to replace the dock connector. As you probably already know it is soldered to the logic board. You will need to unsolder the old one and resolder a new one. Better have good tools, a steady hand and a good magnifying source for that repair. Here is a great video about the repair and the part is available at places like this and many others.