I've not heard the recording, but it looks as though the text is from Sharp's The Cruel Mother "H" set (in English Folk Songs of the Southern Appalachians, sung by Mrs. Mary Gibson at Marion, North Carolina, Sept. 3, 1918:

There was a lady in New York,
   All along little Omy,
She fell in love with her father's clerk,
   All down by the greenwood sidey.

SHe was a-going across the bridge,
She found herself a-growing big.

She leant herself against an oak;
First it bent and then it broke.

She leant herself against a thorn;
And there she had two little babes born.

She carried a penknife both keen and sharp;
She pierced those little babes to the heart.

She burried them under a bunch of rue;
She prayed to the Lord they'd never come to.

When she was a-walking across the porch,
She saw two little babes at play.

O babes, O babes, if you were mine,
I'd dress you up in silk so fine.

Mother, O mother, when we were yours
You neither gave us coarse nor fine.

Mother, O mother, for our sakes
You'll always carry the keys of Hell's gates.

--

Several of the other versions use "All alone and a-loney".

I'll post the tune in a moment.

~ Becky in Tucson