  This is the third part in a series of papers developing a tensor product
theory for modules for a vertex operator algebra. The goal of this theory is to
construct a ``vertex tensor category'' structure on the category of modules for
a suitable vertex operator algebra. The notion of vertex tensor category is
essentially a ``complex analogue'' of the notion of symmetric tensor category,
and in fact a vertex tensor category produces a braided tensor category in a
natural way. In this paper, we focus on a particular element $P(z)$ of a
certain moduli space of three-punctured Riemann spheres; in general, every
element of this moduli space will give rise to a notion of tensor product, and
one must consider all these notions in order to construct a vertex tensor
category. Here we present the fundamental properties of the $P(z)$-tensor
product of two modules for a vertex operator algebra. We give two constructions
of a $P(z)$-tensor product, using the results, established in Parts I and II of
this series, for a certain other element of the moduli space. The definitions
and results in Part I (hep-th/9309076, which has been replaced by a new version
with a greatly expanded introduction and updated references) and Part II
(hep-th/9309159) are recalled.
