I decided to put the English translation here following the Italian. Have found my Italian dictionary so there are no more question marks.

GESU BAMBINO - translation

1. In a humble cabin,
In cold and poverty,
was born the little holy child
that the world will worship.

CHORUS:
"Hosanna, Hosanna" they sing
With jubilant heart,
Your shepherds and angels,
O king of light and love:
"O come let us adore him
   O come let us adore him
   O come let us adore him,
   Jesus, Redeemer."

2. Oh pretty baby, don't cry,
Don't cry, Redeemer,
Your mama rocking you
Kisses you, O Savior.
[CHORUS]

I love the way Jesus is addressed both as a crying baby who needs his mama to comfort him, and the Redeemer and Savior, in the same verse.

Notes –
"Capanna" can mean "hut," "cabin," "hovel," "shed," or "barn."
"pargolo" means "baby, little child."
"culla" means "cradle", "cullare" means "to rock, as in a cradle",
"Cullando" means "rocking," "Cullandoti" means "rocking you."