The mass-mailing worm, which comes as an attachment with one of 18 subject lines, is now showing up in one in every 300 e-mails sent.
The worm sends itself to every address it finds on an infected machine and also tries to deactivate anti-virus software.
Klez.H takes advantage of a known vulnerability in the Microsoft Outlook or Outlook Express e-mail programs.
Since it was first detected in April, anti-virus firm MessageLabs has spotted 775,000 copies of Klez.H. That amounts to 20,000 copies every day.
Number one
This puts it in the number one spot ahead of the SirCam virus which broke last summer and alarmed security experts by maintaining a rate of growth never seen before.
MessageLabs virus specialist Alex Shipp believes it could signal the beginning of a new trend in virus writing.
"When SirCam broke out we thought that we were seeing one of a kind; there was certainly nothing around to match it," he said.
"However, Klez.H has shown us that highly sustainable viruses will continue to appear."
Difficult to spot
How to avoid Klez.H
Update your anti-virus software
Make sure mail software is updated
Be extremely wary of any attachments
Klez.H is a variant of the first Klez virus which appeared in December 2001.
Another variant released by the writer on the same day as Klez.H had hardly any impact.
"Klez.H is able to select random names from address books to use as the sender address and also creates a large range of subject, text and attachment names, making it difficult to identify and track," said Mr Shipp.