  In this work we study modifications in the Aharonov-Bohm effect for
relativistic spin 1/2 particles due to the noncommutativity of spacetime in $2
+ 1$ dimensions. The noncommutativity gives rise to a correction to the
Aharonov-Bohm potential which is highly singular at the origin, producing
divergences in a perturbative expansion around the usual solution of the free
Dirac equation. This problem is surmounted by using a perturbative expansion
around the exact solution of the \textit{commutative} Aharonov-Bohm problem. We
calculate, in this setting, the scattering amplitude and the corrections to the
differential and total cross sections for a spin 1/2 particle, in the
small-flux limit.
