I wonder if the web page has been changed since you posted your message.

The song I heard in the video is LEANING ON THE EVERLASTING ARMS. (The lyrics are in the DT - click the link.)

Strictly speaking, it is not a "traditional" song; it has known writers: Rev. E. A Hoffman and A. J. Showalter.

You can view the sheet music at Google Books, in Sowing and Reaping by J H Kurzenknabe, 1889.

The video is a scene from "Night of the Hunter" (1955), where it is being sung by Robert Mitchum.