Grant, thank-you for the info. Given there is no trickle power it is likely the power supply. However, it would be nice to know if a failed logic board can still illuminate the diagnostic LEDs. At this point it is difficult to tell if it is power supply, logic board, or both. Can a failed logic board still illuminate the diagnostic LEDs? If a failed logic board cannot illuminate diagnostic LEDs and no LEDs are lit, then it it may be more likely the logic board is the problem.
FWIW, the computer is used regularly and the power failed occurred yesterday while powered on. I did not notice any failure symptoms as I was using a different computer at the time.
I tried repowering many ways. With old battery removed. With new replacement battery. Unplugging power cord...waiting...etc, etc. following the Apple manual and other power related threads. In none of the attempts were any sounds heard (e.g., power switch relay) and pressing the diagnostic LED button each time resulted in no LEDs illuminating.
After about 6 hours of reading related threads and articles, I still haven't come across any info stating how to diagnose further when no diagnostic LEDs illuminate (including Apple's own manual). However, given Grant's comments, I'm inclined to think the power supply has failed.
For anyone replacing a Mac Pro 1.1 power supply, be advised: You have to match part of the serial number as well as the part number of the replacament power supply. Part number is usually: XXX-XXXX. The serial number is: XXXXXXXXEEEX. The last 3 of 4 letters or numbers (the EEE code) is the version number of the PSU. These letters/numbers must match or the power supply may not correctly manage the cooling fans.