  The basic objects of the ADHM construction are reformulated in terms of
elements of the $A_{\theta}(R^4)$ algebra of the noncommutative $R_{\theta}^4$
space. This new formulation of the ADHM construction makes possible the
explicit calculus of the U(2) instanton number which is shown to be the product
of a trace of finite rank projector of the Fock representation space of the
algebra $A_{\theta}(R^4)$ times a noncommutative version of the winding number.
