In this thread Top 100 hits of 1950s they said that it was #34 for the year 1958, and sung by Little Anthony.

Assuming it is the same one.

And....

TEARS ON MY PILLOWSung by Little Anthony and the Imperials

You don't remember me, but I remember you
't was not so long ago, you broke my heart in two
Tears on my pillow, pain in my heart, caused by you, you
 
If we could start anew, I wouldn't hesitate
I'd gladly take you back, and tempt the hand of fate
Tears on my pillow, pain in my heart, caused by you,
 
Love is not a gadget, love is not a toy
When you find the one you love, she'll fill your heart with joy
 
If we could start anew, I wouldn't hesitate
I'd gladly take you back, and tempt the hands of fate
Tears on my pillow, pain in my heart, caused by you, you
Oh wo wo wo you you
FADE: wo oh oh oh oh oh oh oh oh