That works great to replace all spaces. However, as a casual observer, I'm not sure this is what was asked. Were all the spaces to be replaced with the non-breaking spaces, or just those after the first one, leaving one space without the replacement. So:
"the lazy brown fox jumped..."
But I would like to expand the question a bit. I would like to see all spaces removed with the exception of just one separating the words, and then have that replace with the non-breaking space. So:
"the lazy brown fox jumped..."
—Brad
"A little yeast leavens the whole dough."