There is no English translation. One book in English which covers most of the material is Akhiezer's book MR1054205. Another is Whittaker Watson. Course of modern analysis.
EDIT. There are very many books covering the basic theory of elliptic functions, old ones and new ones. But I don't know the book which could replace Hurwitz (It is actually three books usually printed under one cover and called Hurwitz-Courant.) There is an excellent Russian translation (edited by Evgrafov), and an expanded modern German version by Rohrl. But there is no English version.