The other song (Grandma had terrific taste, didn't she?)

That's "Ay, Maricruz," by Joaquín "Quinito" Valverde Sanjuán, son of Joaquín Valverde Durán, also a composer. This haunting performance is by the singer Imperio Argentina. Here is a link to the full performance on YouTube:

http://www.youtube.com/watch?v=IGjRL_Dqqm4