For a non-spinning black hole my answer was correct. But as my friend 'relativityguy' points out spinning black holes are real and non-spinning are probable not. The Schwarzchild solution, for spherically symmetric non-spinning bodies of mass, was the first solution to Einstein's equations (first metric). Its good even for objects with small angular speed but a different solution is needed when the black hole is spinning very fast. Follows is what relativityguy said.
"Since you were interested in the behavior of the probe to remote observer clock behavior near r = 0 I
should bring out the point. If you have no charge or spin then you don't have a hole that is physically
realistic, dt/dt would hypothetically diverge at r = 0 as you've suggested. If you add some charge then
you still don't have a realistic hole but in this case the clock never reaches r = 0. If you then add some
spin, you will have a realistic hole at least at the nonquantum scales, but now in the cases that the probe
reaches r = 0 dt/dt will be finite at the origin."
Here is relativityguy's site 'Modern Relativity'.
http://home.aol.com/zcphysicsms/modernrelativity.htm