Yes, it's been solved for a long time. It's solved with limits and the calculus: an infinite sum with continually diminishing components converges to a finite number.
E.g., (I can't use math symbols on my work computer, so E = sigma)
(At infinity, where n = 1) E 1/2^n = {1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...} = 1
In other words, it doesn't really take infinite time to cross the space because it only takes some finite number to cross the infinite divides.
Zeno has several paradoxes, and there are several ways of interpreting this set of paradoxes. This mathematical approach solves but one interpretation of what the paradoxes represent. Thought of as a meditation on mysticism, as Zeno may have intended, since it was asserted as part of a proof for Parmenides' claim that "all is one" the paradox is not necessarily about the impossibility of movement but a paradox on the nature of the 'existence' of things. In that sense, it remains unsolved. Perhaps it is not meant to be solved.