i have studied spanish and the mexican structure in both mexico and spanish, and after listening to the joan baez version of the song a good many times, i am pretty sure that these are the lyrics. . . :
Pajarillo barranqueño.
Pajarillo barranqueño.
!Qué bonitos ojos tienes!
Lástima que tengan dueño

?Quié es ese pajarillo
cantando en aquella lima?
Anda y díle que no cante,
Qué mi corazón lastima.

Little bird, little bird,
Little cliff swallow,
What pretty eyes you have!
Pity that they have an owner.

Who is that little bird
Singing in that lime tree?
Go tell him not to sing,
It wounds my heart.

in some versions, however, the last two verses are also in a spanish/mexican mix