The site, Onebillionplusyou.com, is not yet live – and isn't scheduled to be up until Monday. However, anyone interested in Firefox's progress can see a live twitter feed that is tracking downloads.
Mozilla is estimating about 20 downloads per second worldwide, with a little under 1 million to go before they reach that 1 billion mark. At that rate they'll have reached it before or during this weekend. While this in no way reflects the actual number of Firefox users worldwide, it does indicate the high level of demand people have for the “alternative” browser.
It was only about a year and a half ago that Firefox had managed reach 500 million downloads, indicating that in more recent times demand for the browser has increased tremendously.