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asaris is sort of right. Dr. Hawking was in fact right, but asaris' interpretation of his work is a bit misleading.
The Big Bang was a singularity, just like any other black hole out there; the key difference is that the Big Bang singularity contained all the matter in the universe and, for some reason, it exploded. The why of it is unknown and likely can never be known, since it's outside time.
But the Universe still isn't likely to collapse inwards on itself.
What's being discussed here is generally referred to as the Big Crunch theory. It's one of several models describing the end of the Universe. The Big Crunch depends on the idea that even though the Universe is currently expanding, gravity will eventually catch up with that expansion and cause the Universe to slow down and eventually collapse back inwards.
The other two models that were dominant in the field of cosmology when Dr. Hawking wrote A Brief History of Time posited that the Universe would expand at a constant rate or that it would expand at a perpetually slower rate, but that gravity would not ever be sufficient to halt the expansion. Both of these models lead to one of two ends, being either the Big Freeze (when the heat energy of the Universe is so far spread out that it's impossibe to sustain life) or heat death (when entropy causes the Universe to become uniform in nature, again making life impossible).
These theories are really cool and all, but there's only one problem; all observations since then seem to point to the universe expanding at an increasing rate; in other words, instead of expanding steadily or slowing down, the expansion is actually accelerating. This makes the idea of a Big Crunch highly unlikely, but those doomsday theorists need not fret, because odds are the Universe will one day be unable to support life in any case. This is unavoidable and we cannot escape it (since we are a part of the Universe and necessarily contained within it).
Don't let the Big Freeze or the Big Crunch keep you up at night, though. Our sun will die billions of years before any such scenario comes to pass.
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I wake up in the morning more tired than before I slept
I get through cryin' and I'm sadder than before I wept
I get through thinkin' now, and the thoughts have left my head
I get through speakin' and I can't remember, not a word that I said
- Ben Harper, Show Me A Little Shame
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