"Slept like a baby"

[Bapopik at AOL.COM]

"I slept like a baby.  I woke up in the middle of the night and demanded breastfeeding."
--some comedian (I forget which one)

   One member of my group woke up early and thought we had an earthquake last night.  Another group member woke up to the sound of the church bells at 6 a.m.  I "slept like a baby."
   Why is the saying "slept like a baby"?  Did we cover this before?
   I don't have the OED handy.  Maybe someone from the OED can straighten this out.