This is a WXP computer that starts from the "off" state in about a minute, however, a warm restart takes 15 minutes.

I have defragged, emptied recycle, purged temporary files, deleted the prefetch folder, disabled unneeded startups.

How do I fix this?

If the only difference is between cold start and hot start you would have to consider this is an overheating issue. Go into your bios and check out the temps on your RAM or google Super-pi and download and test.
and, go to the following link and download a HDD temperature test http://wareseeker.com/System/hdd-temperature-1.4.206.zip/308318

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This computer has a temperature monitoring screen built in. CPU runs at about 30-degrees centigrade and the HDD at about 20 degrees centigrade.
I did a "restart" from a running state, which took 15 minutes, then, immediately did a complete shutdown, waited a few seconds and did a restart, which took about a minute.

so the problem isn't overheating, it's something else that is happening during the boot process. It acts like the computer is doing a scan of some kind.

About the only difference between a restart and a start after a shutdown is that the power supply does not get cycled. MB, mem, cpu... all retain power. Software-wise, everything stops and starts anew.
Where exactly in the startup cycle is the extra time being taken? Try running procmon.exe [from winternals] - enable boot logging, reset the filters, and restart the machine. Examine each type of activity [reg , file, network, process ]. See where the time goes.
Malware staying live in memory...? I don't know... try Autoruns from winternals. Run gmer.exe to check for rootkits.
You leave open a lot of possibilities.
http://technet.microsoft.com/en-au/sysinternals/default.aspx
[chkdsk pretty much announces itself when it runs]

Jezza70 & Gerbil: thanks for the input. Will give it a try tonight when I visit my friend who owns the problem computer.

In the meantime, here's another development.
1.) The first time we monkeyed with the Prefetch folder, we renamed it to "Prefetch-Old".
2.) Then we did a total restart and got up and running in a minute.
3,) The next total restart took us back to the 15-minute startup.
4.) We found that the prefetch folder was totally back.
5.) We then sent the folder to the recycle bin and was able to do a total shut-down and 1-minute restart. The warm restart was still a 15-minute wait.
6,) Today, my friend went to do a cold boot and it took 15-minutes again.

The computer goes through the startup process, including the Windows logo screen with the progress bar, then the blue screen with cursor. Next, except for the 15-minute delay, would be the login screen with userid and password required.

I've had a similar problem on a company computer before.

It ended up being a driver issue. We were using some old ID scanners that were built for Windows 98 and the drivers were terribly ported to XP by the sales guy (they were really cheap).

So every time the computer was rebooted, logged off, etc, the drivers would hang up the system and you would have to go into Task Manager and kill the process in order to logoff or reboot (unless you wanted to wait several minutes for the logoff/reboot).

Not sure if that has anything to do with your problem, I'm just saying there's a possibility that it could be driver related.

donkey_pasta: Thanks for the input. This computer was behaving fine until I started trying to network it (XP) to a new computer (W7). XP & W7 aren't easy to network. I have no idea what I changed to cause this problem.

So it is not softwares that are being prefetched. [And yes, prefetch folder entries would be rebuilt with entries for softwares you used during the sessions after deletion]
Try using selective startup via msconfig to isolate the group causing the problem. I could type all day, but M$ tell you how: http://support.microsoft.com/kb/310353
Restore to before the networking experiment?

I went the MSConfig selective startup route per the instructions at http://support.microsoft.com/kb/310353 and on the first restart, things went well. Following that, the computer went back to the slow restart routine. I then did a Windows repair and now the machine starts okay from a totally off state but a warm restart still takes forever.

So it appears that the problem is not memory or the power supply, since the computer starts fine after a total shut down. It is just the restart process that takes forever.

During the reboot cycle how long does it take to turn off compared to the time it takes to boot up?

Try unchecking all the items in msconfig startup and see if that does it.

Also, I've see some system services be more aggressive than others -- so even if you deselected it in msconfig something may be causing it to start.

I'm really curious to see if it takes a really long time to shut off during the reboot cycle or if it takes a really long time to boot up. My guess is it takes a really long time to shut off and if that is the case then there is a program hanging up.

Do you get any kind of error messages or messages asking if you want to "end 'blank' program now" when you reboot?

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