If this is our future [he was discussing the end of the big bang's expansion, which he thought might result in a contraction], it presumably also is our past. The present expanding universe would be only the phase following the last contraction and bounce. (Indeed, in their 1965 paper on the cosmic microwave radiation background, Dicke, Peebles, Roll, and Wilkinson assumed that there was a previous complete phase of cosmic expansion and contraction, and they argued that the universe must have contracted enough to raise the temperature to at least ten thousand million degrees in order to break up the heavy elements formed in the previous phase.) Looking farther back, we can imagine an endless cycle of expansion and contraction stretching into the infinite past, with no beginning whatever. [Underling added by PH.]That's from his last chapter: Epilogue: The Prospect Ahead, which is on page 153 of my copy.
That's a description of a cyclical closed Universe; it made sense when Weinberg wrote "The First Three Minutes" because at that time we had no definitive data on whether or not the Universe was open, closed, or flat.
Since that time, however, the data have pretty much shut the door on the closed Universe. The W-MAP data in particular indicate the matter density of the Universe is exquisitely close, if not exactly equal, to the critical value, which corresponds with a flat Universe that never collapses, especially when the accelerating expansion is considered.