  A {\em hereditarily indecomposable (or H.I.)} Banach space is an infinite
dimensional Banach space such that no subspace can be written as the
topological sum of two infinite dimensional subspaces. As an easy consequence,
no such space can contain an unconditional basic sequence. This notion was
first introduced in 1993 by T.Gowers and B.Maurey, who constructed the first
known example of a hereditarily indecomposable space. Gowers-Maurey space is
reflexive, however it is not uniformly convex. In this article, we provide an
example of a uniformly convex hereditarily indecomposable space, using similar
methods as Gowers and Maurey, and using the theory of complex interpolation for
a family of Banach spaces of Coifman, Cwikel, Rochberg, Sagher and Weiss.
