Tom/Barb wrote:
Can't be the lift pump, because the engine shuts down when the lift pump is turned off by the computer.
I'd change the engine fuel filter.  first.
If that didn't fix it, I'd trouble shoot buy placing a temporary grounding strap from the ECM to a known good ground. If that fixes it, remove the ECM clean the mounting pads to provide a good ground.
No offence but thats not accurate. The HPFP can, depending on exactly where the lift pump fails, often draw fuel through the dead LP. 
Also, the engine shuts down when the computer stops firing the electronic injectors. This is different from older models that just stopped fuel flow - which was not done at the fuel pump, but at the inj. pump.