By auto-antenna, I assume this is the kind that goes up and down by itself, when you turn the key to "on". I'm assuming you've verified the antenna is going up and down as it is supposed to. And that there is continuity on both the signal and shields from one end to the other. If so, then the splice is the problem.
Connecting one piece of coax to another is usually done by putting a connector on each of the wires, then connecting the connectors together. I've done this inexpensively w/the rf connectors used for tv antennas, "F connectors" I think they call them. Radio Shack sells these, and this is fairly easy to do, if you're up to doing it yourself. You need one female, and one male type. Or two of the same type with a barrell adapter.
F connectors are for 75 ohm cable. They should work fine for you as the run is fairly short, but there may be a different type of connector typically used for car antennas, as the cable may be 50 ohm instead. Maybe Google knows.
Replacing the entire line with a new cable would work too.