That's "Fisher Row" aka "Upstairs, Downstairs":

Now she maun mount the cutty stool,
And I maun pay the pillar
That's the way the poor folks dae
Because they hae nae siller

She gets to sit on the Stool of Repentance (whatever that is)--he gets put in the stocks. Clearly an unpleasant public consequence for getting caught out, but not in the same league as being murdered by your own family.

And there's a clear suggestion that they could buy their way out of the predicament--which also suggests that the real problem, as illuminated in Nancy's interesting post, is that they can't afford to get married and set up housekeeping. (And a lack of birth control options--the song also pointedly remarks:

"Had her apron bidden doon, the kirk would ne'er ha kent it
Noo the word's gang roond the toon, I fear I canna mend it.

One of my favorites to sing, as is "Gaberlunzie Man" for three reasons:

1. they don't get caught
2. I just love singing the phrase "Gaberlunzie Man"
3. I am

   Willie-O