No it never says seperation. But it sure makes it clear that the government cannot show any kind of preference of one religion over another, or non-religion. But it was Thomas Jefferson who said there would be a "wall of seperation between church and state" in a letter to a baptist group.
Believing with you that religion is a matter which lies solely between Man & his God, that he owes account to none other for his faith or his worship, that the legitimate powers of government reach actions only, & not opinions, I contemplate with sovereign reverence that act of the whole American people which declared that their legislature should "make no law respecting an establishment of religion, or prohibiting the free exercise thereof," thus building a wall of separation between Church & State.
James Madison, a primary author of the Bill of Rights, stated something quite similar: practical distinction between Religion and Civil Government is essential to the purity of both, and as guaranteed by the Constitution of the United States
President John Tyler would later write: The United States have adventured upon a great and noble experiment, which is believed to have been hazarded in the absence of all previous precedent-that of total separation of Church and State. No religious establishment by law exists among us. The conscience is left free from all restraint and each is permitted to worship his Maker after his own judgement. The offices of the Government are open alike to all. No tithes are levied to support an established Hierarchy, nor is the fallible judgement of man set up as the sure and infallible creed of faith. The Mahommedan, if he will to come among us would have the privilege guaranteed to him by the constitution to worship according to the Koran; and the East Indian might erect a shrine to Brahma if it so pleased him. Such is the spirit of toleration inculcated by our political Institutions . . . . The Hebrew persecuted and down trodden in other regions takes up his abode among us with none to make him afraid . . . . and the Aegis of the Government is over him to defend and protect him. Such is the great experiment which we have cried, and such are the happy fruits which have resulted from it; our system of free government would be imperfect without it."
I agree with you the exact phrase of "serparation of church and state" is not in the Constitution but you can see where it does come from.