Other equally official sources specify the other way so it all depends on which guide you follow as to which way you write it. The most important thing for any site is to pick one or the other and use it consistently for all such references on the site.
My thoughts as well. Never use the separated version myself.
One issue I have with the AP Stylebook is that it uses Web site and Web page but at the same time uses webcam, webcast, and webmaster. So not only does it have one word and two word differences, there's a capitalization difference. So while I go by those for work, personally I find myself doing the one word version of everything.
AP Style suggests Web site as the correct spelling - which is then accepted by most media. The reason the accepted version of Web site is different than webcam, etc. is because Web site and Web page are considered exceptions from Webster's spelling version as the others are not.
I did a snap survey around the office (me and 9 technophobes) and there was a 50/50 split between the 2 options. Although the guys who went with website as one word all said it was because that's the way they usually see it online.