Well, I found "Those Dark Eyes" by Jeanie West, and I think it's the right one. It's hard to make out the lyrics. this is about as close as I can get.

THOSE DARK EYES
(traditional?)

The evening sun was sinking low
When dark eyes murmured I must go
Not one moment would he wait
He kissed my cheek and left the gate

Now, it's those dark eyes I love so well
Now it's those dark eyes I long to see
And now I cry for those Dark Eyes
A stranger they have grown to be

The last we met was on the street
I bowed my head and said godspeed
There was another by my side
Who's begging me to be her (???) bride
CHORUS

There beneath the churchyard grave
They laid my own Dark Eyes away.
Ran from everything I need
And let those Dark Eyes wait for me
CHORUS

Transcribed by ear from the Jeanie West CD, Country Bluegrass (LP was Roaming the Blue Ridge with Jeanie West)
-Joe Offer-

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As noted above, The Dixon Brothers also recorded a song called "Dark Eyes."