Help with a 1965 Mustang with a 200 inline 6, will not stay running.

Did you change the fuel filters under the car and on the side of the carburetor and take apart the fuel lines and blow it out

Is your choke manual, electric, or does the heater hose run up next to it in order to warm it up, and is it adjusted correctly? Is the Carb new or rebuilt? What kind of carburetor is it? IF it is new it should come with some instructions on how to adjust the choke and what setting it should be. Your air fuel mixture is probably messed up. Either to much gas or not enough. When it starts does black smoke come out of the tail pipe, if so it is too rich (too much gas). It could be caused by the choke or jets or if equipped the power valve may be bad. Sometimes "rebuilt" carbs can still have bad power valves. A power valve essentially squirts additional fuel into the carburetor's venturi immediately as you depress the gas pedal, it does this in order to prevent a lean mix causing the engine to stall. If the valve is not the right size or the rubber diaphragm inside the valve itself is ruptured it can cause engines to stall out when you hit the gas. I know where it would be on a 4 barrel, it is directly in from of the throttle linkage near the top of the carb itself. It also has a linkage that is attached to the throttle linkage. You may want to replace your fuel filter as well. Oh and BTW, what is a the "button" that you replaced along with the points?

adjust your timing try turning dist.and make sure point gap is set

I'm an old guy, i use to have this problem on these old cars when coiled controlled chokes wouldnt work anymore. If all else fails if you can find it buy a hand controlled choke adapter. Than play with it when it chokes, if this keeps it running you can just run it like this or than know your carb isn't adjusted right and go from there.

Rebuild the carb