Doesn't Einstein's theory of Relativity make impossible the Big Bang from an initial Black Hole Singularity?

The Big Bang is supposed to have occurred from an initial singularity containing all matter in the Universe in an extremely compressed space. Hence the curvature of space-time would be infinite at the time of the Big Bang, in the area in which all the matter of the Universe was compressed. So, how could the Big Bang possibly have occurred? By definition, in a Singularity, time ceases to pass, and no events can occur. And, the Big Bang was supposed to have been one Hell of an event, wasn't it? So, on the surface at least, it would seem that we can have Einstein's Theory of Relativity, or, we can have the Big Bang, but not both, because they are mutually contradictory. What is the explanation for this? I assume there must be one, of course.