Nutty: Yes, it does look like the same song! But the version used in the play must have been "folk-processed" a bit.

From The Lester S. Levy Collection of Sheet Music:

THE ANCHOR'S WEIGHED
Braham.
"Sung by Mr. Geo. Simpson"
[no date]

1. The tear fell gently from her eye
When last we parted on the shore.
My bosom beat with many a sigh
To think I ne'er might see her more,
To think I ne'er might see her more.
"Dear youth," she cried, "and canst thou haste away,
My heart will break; a little moment stay.
Alas! I cannot, I cannot part from thee.
The anchor's weigh'd. The anchor's weigh'd.
Farewell! Farewell! Remember me.

2. "Weep not, my love," I trembling said.
"Doubt not a constant heart like mine.
I ne'er can meet another maid
Whose charms can fix my heart like thine!
Whose charms can fix my heart like thine!"
"Go then," she cried, "but let thy constant mind
Oft think of her you leave in tears behind."
"Dear maid, this last embrace my pledge shall be.
The anchor's weigh'd. The anchor's weigh'd.
Farewell! Farewell! Remember me."