Nope. He remains HIV positive and has not (yet) progressed into full-blown AIDS. Keep in mind that when diagnosed he was (and likely still is) in otherwise extremely good heath, plus he had boatloads of money to spend on early promising treatments. If anyone is likely to be a long-term HIV survivor, it’s him, but unless something else gets him first, I expect he’ll eventually succumb.
I haven’t researched this story, but the linked story does not indicate which tests were used. In the US at least, an initial positive is confirmed with a second test called the Western Blot. Can both be falsely positive, yes, but extremely rarely. These tests, though, only confirm the presence of HIV antibodies, not the actual viral activity (see below).
HIV viral activity is measured with a test called the “viral load” http://www.labtestsonline.org/understanding/analytes/viral_load/test.html , which detects quantitatively the level of HIV viral activity (not just antibodies). Even if a standard viral load test reports that the virus in undetectable, this doesn’t indicate cure either because there is a lower limit of detection. Ultrasensitive measures of viral load would be required to suggest a “cure”, and even then, repeated measurements in the future would be necessary to prove a “cure”.
Natural immunity could develop if an individual’s immune system was uniquely capable of destroying the virus when exposed, but so far nobody knows who those folks might be.