2006 V6 Mustang : stops running?

Does it hesitate at all? Do the RPM's drop slowly? Has the engine surged for no reason?

if its over 125,000 miles, it could be the fuel pump. Not from personal experience but my 2008 V6 has 115,000 miles on it with plenty of preventive maintenance. Fuel pump is a likely to go bad eventually on mine.

 

I'd replace the entire fuel pump assembly, not just the pump.
There is a non-replaceable valve inside the pump housing, often referred to as a regulator.
Mine was doing the same thing as yours, so I got a fuel pressure test gauge, and it barely registered any pressure.
Replaced the pump, no change.
Also, sometimes the plastic hoses inside the fuel pump bucket develop leaks from years of heat cycles, they harden and get brittle.
I read about the internal valve, replaced the whole ''bucket'' the pump is installed in, and now the car runs fine, 40 psi.

 

This is he valve inside the fuel pump bucket, it is completely blocked, both ways.
Along with the fuel rail pressure sensor (FRPS) and fuel pump control module, (FPCM) it regulates fuel pressure.
Also, note the plastic fuel line, it is supposed to be white. Disregard the date stamp.

 

I would get the entire assembly, not just the pump.
If you replace only the pump, and that valve is bad, you will have to pull the fuel tank out again, as the pump is inside the tank.

 

Do what Pete fender says.
Maybe a fuel filter change is a good idea too