  In this work we give a method for constructing a one-parameter family of
complete CMC-1 (i.e. constant mean curvature 1) surfaces in hyperbolic 3-space
that correspond to a given complete minimal surface with finite total curvature
in Euclidean 3-space. We show that this one-parameter family of surfaces with
the same symmetry properties exists for all given minimal surfaces satisfying
certain conditions. The surfaces we construct in this paper are irreducible,
and in the process of showing this, we also prove some results about the
reducibility of surfaces.
  Furthermore, in the case that the surfaces are of genus 0, we are able to
make some estimates on the range of the parameter for the one-parameter family.
