  Consider a domain D in R^3 which is convex (possibly all R^3) or which is
smooth and bounded. Given any open surface M, we prove that there exists a
complete, proper minimal immersion f : M --> D. Moreover, if D is smooth and
bounded, then we prove that the immersion f : M --> D can be chosen so that the
limit sets of distinct ends of M are disjoint connected compact sets in the
boundary of D.
