We prove that every open Riemann surface M is the complex structure of a complete surface of constant mean curvature 1 (CMC-1) in the three-dimensional hyperbolic space ℍ3. We go further and establish a jet interpolation theorem for complete conformal CMC-1 immersions M → ℍ3. As a consequence, we show the existence of complete densely immersed CMC-1 surfaces in ℍ3 with arbitrary complex structure. We obtain these results as application of a uniform approximation theorem with jet interpolation for holomorphic null curves in ℂ2 × ℂ∗ which is also established in this paper. [ABSTRACT FROM AUTHOR]