As far as I know, it is only a conjecture that string theory converges to all orders. Here is a review article with some discussion of the state of knowledge about this as of 2003:
http://arxiv.org/abs/hep-th/0303185v2 See p. 34, where he states that the conjecture has not been proved. I don't think the conjecture had been proved as of 2006, which was when Smolin's The Trouble with Physics was published; on pp. 278-281 of that book he states that finiteness to all orders still had not been proved.
As far as the reasons why it might be expected to be finite, the arxiv paper says,
"There are intuitive arguments that suggest that ultraviolet divergences of the kind that plague conven-
tional quantum field theory cannot occur in string theory. The main reason is that the
interactions of strings involve the breaking and joining of strings and these do not take
place at points. However, a string theory can fail to be consistent for other reasons.
There may be infrared divergences, or ambiguities in the definition of the amplitudes,
there can be anomalies in the action of the lorentz boosts, or the theory may fail to be
unitary."
