We who hold public office are enjoined by our Constitution against enacting laws to tell the people when or where or how to pray.
All our experience and all our knowledge proves that injunction is good. for, if government could ordain the people’s prayers, government could also ordain its own worship — and that must never be.
The separation of church and state has served our freedom well because men of state have not separated themselves from church and faith and prayer.

Lyndon B. Johnson (1908-1973) American politician, educator, US President (1963-69)

Speech (1964-02-05), Presidential Prayer Breakfast, Washington, D.C.
    (Source)

This was at the 12th Annual Presidential Prayer Breakfast.

In the Proceedings of the Illinois State AFL-CIO Convention (1968), there is (a) a reference to a note that the state president of the AFL/CIO, Reuben G. Soderstrom, attending the 16th such Prayer Breakfast, and then (b) a passage on the next page "U. S. President Lyndon B. Johnson's statement to a tremendous audience contained the following comment:"

Our Constitution separates church and state. We know that separation is a source of our system's strength, but the conscience of our nation does not call for separation between men of state and faith in the Supreme Being.

Johnson does not appear to have included this text in his speech at the 16th Presidential Prayer Breakfast, nor does he appear to have gone to the 1968 Illinois AFL/CIO convention. Is this an odd paraphrase of the comments from four years earlier? Did Johnson speak the above in another venue that was also quoted in the Illinois AFL/CIO Convention proceedings? Is this paraphrase actually what he said in 1964, regardless of the written record of his comments?

While that shorter quote, or further paraphrases of it, are easy to find in quotation collections online, I can find no citation associated with it.