The exact words are "Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof" - the second part guarantees the freedom of religion as you mention, but the first part does guarantee what you might accurately describe as separation of church and state, by forbidding Congress (and, as the Supreme Court has made clear afterwards, other levels of government) to "establish" any religion.
Mind you, I'm on the side who is inclined to agree with this decision, though not wholeheartedly. The Constitution does absolutely ban the city of Greece from establishing Christianity as the town's official religion, the conservative justices' opinions were very clear on that. They just ruled, rightly I think, that Greece's actual practice doesn't constitute "establishing" Christianity, and that while it's not the optimal way of handling things (Alito made several remarks about how Greece's policy was certainly not a "best practice", it's not so bad as to violate the First Amendment and require striking down.