Quiz: The Probability of Having a Disease

A new super-resistent virus has emerged in Farawayistan. The Ministry of Health estimate 1 out of 1000 to be infected. After immense efforts, a very accurate test has been developed. If the test subject is infected, 99.9% of the time the test will come out positive. But if the subject is not infected, 1 in 1000 times the test will yield a false positive result.

A random person is tested, and the test comes out positive. What's the probability that he's infected?

Please, answer in the comments.