If you really mean you only want to count first-time visitors, you'll have
to place a cookie so you can recognize "veterans."
Then, your index.htm page can be a flash screen, something nice, that says
"Click here to enter." No hit counter. It checks the visitor to see if
you've placed a cookie with her.
If the visitor has no cookie from you, you direct her to a "home page" with
a counter. Otherwise, she is sent to an identical "home page" with no
counter. It only requires that you create and maintain two home pages.
Otherwise, your website can remain the same.
And you get to learn how to handle cookies.