First thing you do is get yourself a factory service manual if you don't already have one. You can download most of them here, but I see they don't list the '65 Dodge, so a Plymouth version is going to be close enough for this.
MyMopar - Mopar Forums & Information - Service Manuals
The starting system works like this...
Turn the key, it activates the starter relay mounted on the firewall. The starter relay activates the solenoid on the starter. If the car isn't in neutral, the starter relay won't work.
So... Since you've eliminated the starter, battery, and battery cable, you've got three other possibilities.
One is the ignition switch. Second is the starter relay (usual problem) and third is the neutral safety switch (least common problem)
starter relay looks like this:
Bottom right terminal is the wire coming up from the NSS. Since that's been brought up, let's check that first. Disconnect that wire and using multimeter (a cheap Harbor Freight version will do it) check to see if there is continuity to ground when the car is in park and neutral. The alternate test, that doesn't use a multimeter, is to run a jumper wire from that terminal to a good ground and try to start the car.
Now you've probably found that these guys are full of **** about the NSS, so let's go on.
The lower left terminal is the feed from the ignition switch. When the key is turned to "start" position, you should have 12 volts to that switch and you can check that with your multimeter. Alternative is to momentarily connect a jumper wire to a 12 volt source. Happens to be a good one at that terminal on top. If jumper is attached and the engine turns over, the ignition switch (or the wiring to it) is the problem.
Now we check the starter relay... This is probably your problem, but now you know how to check the rest.
The upper two terminals... The top is the battery feed and the one below it is the feed to the starter. You take an old screwdriver and short it between those two terminals. Since you've established that the starter is working, that should turn the engine over. Replace the starter relay.
