What is the song that Marian sings at the piano?

The particular song I am referring to is heard about 7 minutes into the film, as we see Leo brushing his hair in front of the mirror, preparing for dinner. In the background we hear Marian entertaining guests at the piano, singing this song.

Later on the film, we also hear her sing another song, 'Sweet Lass of Richmond Hill.'

If anybody recognises the first song, please let me know. Thanks.