  We consider integrable superstring theory on AdS_3 x S^3 x M^4 where M^4=T^4
or M^4=S^3 x S^1 with generic ratio of the radii of the two 3-spheres. We
compute the one-loop energy of a short folded string spinning in AdS_3 and
rotating in S^3. The computation is performed by world-sheet small spin
perturbation theory as well as by quantizing the classical algebraic curve
characterizing the finite-gap equations. The two methods give equal results up
to regularization contributions that are under control. One important byproduct
of the calculation is the part of the energy which is due to the dressing phase
in the Bethe Ansatz. Remarkably, this contribution E_1^{dressing} turns out to
be independent on the radii ratio. In the M^4=T^4 limit, we discuss how
E_1^{dressing} relates to a recent proposal for the dressing phase tested in
the su(2) sector. We point out some difficulties suggesting that quantization
of the AdS_3 classical finite-gap equations could be subtler than the easier
AdS_5 x S^5 case.
