Any Ideas on Engine Starting Problem?

[musclecar fan]

I live in Los Angeles and have a 69 RS/SS Convertible and am having engine starting problems. Here is what is happening: Sometimes when I go to start the car in the morning, or after I have already driven awhile I am unable to get the car to start-up. I turn the key and I only get is a single "click" which sounds like a relay or the starter solenoid or maybe something else. The engine does not crank over. If I have someone give me a "jump" with their battery I can start the car. My battery is new and is fully charged (I even changed it to make sure it wasn't the battery) and my starter and solenoid are new too (I even changed that again to isolate the issue). I still have the same problem, namely even if I get the car started, once I get to my location and shut the engine off, all I get is the single "click" sound. Someone suggested changing the ignition switch and/or the horn relay. Any ideas on what may be causing this?