Sorry, Wayne, I tried. I remember the song too, but no more lyrics than you. My mother used to sing it except she sang "That" instead of "This". Although in her seventies, she's still alive, so I called her. She didn't remember where she got it and said she might have gotten it from her mother.
I went to my favorite place for old songs which is http://ntl.matrix.com.br/pfilho/html/main_index/
There I found a #4 Hit in 1931 entitled "Little Girl" but they don't have the lyrics up yet. It was performed by Joe Ventuti and featured Harold Arlen. You might try back with them from time to time as they haven't had 1930 thru 1949 on there very long so I'm sure more lyrics will be added.
AS FOR THE REST OF YOU GUYS, WHY DON'T YOU GO THERE AND LOOK AROUND AND SEE IF YOU CAN HELP THEM OUT WITH SOME LYRICS.