As far as I can tell, it's not a film but a stage play or musical revue.

The original 1997 cast recording of "The Irish ... And How They Got That Way" (Varèse Sarabande VSD-5916, 1998) gives the title as THE ANCHOR'S AWEIGH. It is sung by Ciaran Sheehan and Donna Kane. I transcribed this from a few sound samples:

The anchor's aweigh. The anchor's aweigh.
Fare thee well, fare thee well, remember me.

Weep not, my love, I trembling said.
Doubt not a constant heart like [mine?]...

The notes I've been able to find give no origin information.