The original was published in German in a couple of places. A translation was published by the AMS in 1902.
This article has been reprinted in 1976 by the American Mathematical Society (see references).
The AMS Symposium mentioned at the end contains a series of papers on the then-current state of most of the Problems, as well as the problems.
The URL contains the list of problems, and their current status:
Mathematical Developments Arising from Hilbert Problems, volume 28 of Proceedings of Symposia in Pure Mathematics, pages 1--34, Providence, Rhode Island. American Mathematical Society, 1976.
D. Hilbert. Mathematical problems. Lecture delivered before the International Congress of Mathematicians at Paris in 1900. Bulletin of the American Mathematical Society, 8:437--479, 1902.