  We study when an sl(2)-representation extends to a representation of the Witt
and Virasoro algebras. We give a criterion for extendability and apply it to
certain classes of weight sl(2)-modules. For all simple weight sl(2)-modules
and those in any of the abelian Krull-Schmidt categories of weight modules
whose unique simple object is a dense module, we fully characterize which ones
admit extensions, and we obtain explicit expressions for all of them. We also
give partial results in the same direction for the abelian categories of weight
modules which have two and three simple objects.
