Thank you all for your comments. I have reached a decision.
The answer has already been given by Alanrf.
I will quote him:
"trigan,
I just fired up my Windows Me laptop. Guess what ... my ashWevSv.exe gets a "not responding" too. It is working just fine like yours.
Just for clarification and completeness ...
This a WinMe system rebuilt from scratch less than a year ago and seldom used and certainly for nothing more adventurous than listening to BBC radio via Firefox. No other antivirus (software) used but Avast and no firewall (ZA kept turning itself off - so it was removed) but also only ever used behind a hardware router."
Alan says that only Avast anti virus software was installed. So this clearly proves that even if I uninstall Norton System Works completely, the same problem should STILL be there.
So I have decided not to go through all the trouble. Thank you, Alan. You have saved me a lot of trouble.
As I have already said, after I installed the latest version of Avast, I noticed that CRL ALT DEL was showing that ASHWEBSV.EXE was not responding and ASHSERV.EXE was not even listed in the list of running programs !! A month later, I started getting the computer suddenly giving me a blue screen indicating that an exception had occurred and that Avast had caused an invalid page fault . This happened after I had been surfing using Firefox for about an hour or less.
The strange thing is that this used to happen with Norton Anti virus as well when I had it running on startup. This is one reason why I stopped the Norton anti virus from running on startup. I used to run the anti virus program only to perform a monthly scan and then shut it. And I hardly ever experienced a blue screen of death after that.
So unless any of you think it is a really bad idea, I think I will not run Avast anti virus program automatically on startup and will not therefore have the Standard Shield. I will program Task Scheduler to automatically open up ASHAVAST.EXE once every two weeks and do a thorough scan of all my partitions.