I've just had another listen. The verse should start:

  Down by the river Rio Grande

I now think the last word of the penultimate line of the verse is turning, but I'm not sure it fits well with the start of the line; I may have that wrong.

and the last lines of the chorus:

  Señorita, here is my heart.


I've put my current version below.

Mick

---

RIO RITA
(Tierney, McCarthy)


Down by the river Rio Grande
On silver sand,
That's where I found you
And now my heart's at your command
Wants to be near your hand,
Lingering round you.
For love like mine isn't only turning
Give me one sign, tell me you are yearning, learning

Rio Rita,
Life is sweeter, Rita, when you are near.
Rio Rita,
Life's completer, Rita, to have you, here
All I ask from above
Is one day your lips will say I love you
Rio Rita
Señorita, here is my heart.

[Repeat from] All I ask from above ...