_Sing A Song With Me_ was written by Dame Barbara Cook's friend and songwriter Wally Harper (1941 - 2004); the words are credited to Paul Zakrzewski.  

John Yogis writes: "He wrote it as a theme song for her Carnegie Hall concert 1975, and it also appears on at least one other album on which they collaborated."

It appears on her 1975 album, "Barbara Cook at Carnegie Hall."

Here's a slightly better copy of it, another quite good recording, and probably one of the last times she performed it.
* https://www.youtube.com/watch?v=ldOEKwYzZnY
* https://www.youtube.com/watch?v=z-TpRD2ic2g
* https://www.youtube.com/watch?v=0Z0igIKO8JE