The US Constitution does not require separation of church and state.

First Amendment says "Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof;  
Establishment is an old fashioned word that we don't use hardly at all anymore in America.  Back in 1789 establishment was enjoyed by the Church of England in England.  You had to be a member of the Church of England to receive a commission in the Royal Army or the Royal Navy.  Catholics, Quakers and others were banned.  You had to be a Church of England member to be crowned as king, or ruling queen.  And a number of other goodies were reserved for Church of England members only.  In the Commonwealth of Massachusetts the Congregational Church (Puritan's they had been called for many years) was established.  I'm not exactly sure just what bennies were reserved for Congregationalists in colonial Massachusetts, but Massachusetts did not "disestablish" the Congregational church until maybe 1808 if memory serves. 
   At Constitution time there were a number of large and influential churches doing business in North America.  Congregational, Quaker, Episcopal, Methodist, Catholic, and others.  It was an easy decision on the Founder's part to prevent endless lobbying, back stabbing, and wheeling and dealing by saying that no church will get the bennies of being established.  Massashusetts was required to disestablish the Congregational Church. 
   In short, the establishment clause merely puts all churches on equal ground, no church gets special bennies for their members.  It does not call for separation of church and state, it calls for treating all churches alike.