I think the song you're looking for is Around The World by Nat King Cole. It was the theme tune for the 1956 movie Around The World In 80 Days, although the movie actually used an instrumental version.
Around The World has since been covered by numerous artists including Frank Sinatra, Brenda Lee, Bing Crosby and Jonah Jones.
Lyrics to Around The World by Nat King Cole
Around the world, I've searched for you
I traveled on when hope was gone To keep a rendezvous I know somewhere, sometime, somehow You'd look at me And I would see the smile you're smiling now It might have been in County Down Or in New York In Gay Paree or even London Town No more will I go all around the world For I have found my world in you It might have been in County Down Or in New York In Gay Paree or even London Town No more will I go all around the world For I have found my world in you
