You very well could have multiple problems and combinations of these problems leading you down false twists and turns. You didn't bother to to mention what year model so I don't know whether this was the first year Ma made a slant or the last year. Regardless, it is still decades old, and this means the fuel system, the engine, and the wiring is decades old, and if it has not been very well maintained and even rebuilt there could be serious problems
Depending on where you are, there could be serious corrosion problems in the electrical system. Internal corrosion in the starter relay, cables, and starter. You cannot just stand there and look at it, you have to check it.
Clip a voltmeter on the starter and crank the engine. What does the voltmeter read under cranking conditions?
Remove the battery, haul it to someone with a "carbon pile" tester who knows how to use it and who is NOT interested in simply selling you a new battery and have the battery tested. The battery must be charged before testing, but not immediately before testing. THIS is a carbon pile tester:
Generally speaking you are up against this:
Battery condition
Battery cables and main power distribution to starter, IE main cables
of course the starter itself
Engine condition, IE rings, valves and valves condition
Carb condition, fuel system condition, and condition of the fuel itself.
Condition of ignition system and state of ignition system tune. Just because "stuff is new" does not mean it is "good" that is functional. For example a brand new distributor might have stuck advance weights, or you might have made a mistake setting timing. You might even have made a mistake installing the plug wires
Actual troubles in the system, such as faults in the electrical system. A very good example, is when you twist the key to start, the ONLY power the ignition system gets for start, is a separate circuit known as "IGN2" or the bypass circuit. This is a separate contact in the ign. switch, goes through the bulkhead connector, to the coil + side of the ballast resistor. If this is not getting "hot juice" during cranking, you will not get a good hot starting spark. You could have problems with the bulkhead connector, with the ignition switch connector, or with the switch itself.
READ please this MAD article which points out problems with the bulkhead connector:
http://www.madelectrical.com/electricaltech/amp-gauges.shtml
Do yourself a favor and go here
http://www.forabodiesonly.com/mopar/showthread.php?p=1970088617
and here
http://www.mymopar.com/index.php?pid=31
and download yourself a service manual for your car
And if you are serious about obtaining help here, post the YEAR of your vehicle. Believe it or not, it is important to the rest of us to get some idea about what might be going on!!
You need to get, if you do not have, some basic tools for electrical troubleshooting
A bag of clip leads from Radio Shack. A digi multimeter, and a 12V test lamp.