The First Amendment to the Constitution says that Congress "shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof." It was basically meant to prevent a State church.

The idea of complete separation of Church and State started with a letter from Thomas Jefferson to the Danbury Baptist Association in 1802. Jefferson argued that the First Amendment effectively built a “wall of separation between the church and state.”

Jefferson's view has obviously influenced how we view the issue ever since (as it should), but the idea of complete separation as most see it now is not in the Constitution.