Solved! It's a version recorded - and probably rewritten - by the Appalachian autoharpist (John) Kilby Snow. Here are the lyrics, reproduced from a page put up by a singer called Robin Greenstein:

The Wind and Rain

It was early one morning in the month of May
Oh the wind and the rain
Two lovers went walking on a hot summer's day
A crying the dreadful wind and rain

He said to the lady "won't you marry me"
"And my little wife you'll always be"

Then he knocked her down and he kicked her around
(repeat)

He hit her in the head with a battering ram
(repeat)

He threw her in the river to drown

He watched her as she floated down
(repeat)

She floated on down to the miller's millstream
He watched her as she floated down

The miller fished her out with a long fishing pole
(repeat)

He made fiddle pegs of her long finger bones
(repeat)

He made a fiddle bow of her long curly hair
(repeat)

The only tune that fiddle would play, was
Oh the wind and the rain
The only tune that fiddle would play, was
A crying the dreadful wind and rain

A contemporary review of Kilby Snow's 1969 LP Country Songs and Tunes with Autoharp, written by D.K. Wilgus and published in the Journal of American Folklore 83(329), is informative if a bit sniffy:

His "Wind and Rain" is one of the most interesting ballads recovered recently. The tune and refrain and the stanzas in which a miller fishes a girl from a river and makes a musical instrument of her body belong (as far as we know) to a local tradition of Child 10 ... but the ballad does not tell the tale of sibling rivalry, as does "The Two Sisters." It is instead a "murdered girl type" in which a spurned lover knocks the girl in the head with a battering ram and throws her body into the river. It would seem that a singer, recalling only the conclusion of the ballad, reconstructed the story in terms of the best-known plot in which a girl is murdered. I would suggest "Wexford Girl" (Laws P35) as the model. Kilby Snow is the likely recomposer, as he tells it:

"The first time I ever heard that was my old grandfather [Thomas Snow, a Cherokee Indian also known as Tom Big Bear]... I heard that tune, I don't know, some way or other it just got in my mind. Then the old man died. Went on 'til here I guess just a couple years ago that I thought of that devilish thing. I just happened to be setting there studying about, thinking about the old man ... and that tune come to my mind. So I studied it all up and put it back together as near as I could from remembrance the way that he singed it."

This is illustrative of a number of problems: the textual classification of ballads, the associational relationship of "narrative units," the determining force of a local ballad pattern ("the murdered girl"), and so forth.

I guess one folkie's "problems" are another's "the kind of issues that make the folk process so interesting"...