A quick bit of "googling" later confirmed that it is "bear".
Turns out that "bear" is a very versatile word, while bare is not.
The former is used in the following phrases:
- Bear with me
- Bear in mind
- Bear the pain/suspense/cold
- Bear a grudge
- Bear no ill will
- Bear fruit
- Bear children
- Grin and bear it
- Bring to bear
Bare, on the other hand, just means without clothes.
Given this understanding, it is pretty clear that many of these phrases would have very different meanings if you were to substitute "bare"! "Bare in mind", for example, could mean vacuous while "grin and bare it" would be something that a stripper might do.
Here is a picture to help remind you of the proper spelling for "bear with me" (taken from a TV show called "The Life and Times of Grizzly Adams" back in the 70s):
