  We construct a five-parameter family of gauge-nonequivalent SU(2) instantons
on a noncommutative four sphere $S_\theta^4$ and of topological charge equal to
-1. These instantons are critical points of a gauge functional and satisfy
self-duality equations with respect to a Hodge star operator on forms on
$S_\theta^4$. They are obtained by acting with a twisted conformal symmetry on
a basic instanton canonically associated with a noncommutative instanton bundle
on the sphere. A completeness argument for this family is obtained by means of
index theorems. The dimension of the ``tangent space'' to the moduli space is
computed as the index of a twisted Dirac operator and turns out to be equal to
five, a number that survives deformation.
