need help from auto mechanic

[Ether]

Are there any auto mechanics in CD-land who would be willing to toss out some ideas on what could be the cause of this perplexing problem with my daughter's 1998 Chevy Lumina with 3100 SFI V6 engine?

The starter motor fails to crank under the following sequence of steps:

1) engine dead cold, then
2) start the engine (starts OK), then
3) turn the engine off after running approx 1 minute, then
4) try to start engine again and it will not crank. No clicking, no buzzing, nothing.

I slapped a voltmeter on the battery to see what was happening in step 4, and there is no drop in voltage when turning the key to the cranking position.

I tried two different keys (the keys have a security chip in them) and the problem exists with both keys.

Once the engine is warm (drive, say, 10 miles) the engine will repeatedly re-start with no issues.

The engine also always starts the first time when cold. The problem occurs only when trying to re-start as described above.

Once the engine fails to crank (as described above), I have to wait for a while* and then it will crank (and start) again.

I am at a loss for a working hypothesis. Any ideas what could be wrong?


* I haven't got a handle on the minimum required wait time yet. Three hours seems to be sufficient, but it may be shorter.