  QED based on $\theta$-unexpanded noncomutative space-time in contrast with
the noncommutative QED based on $\theta$-expanded U(1) gauge theory via the
Seiberg-Witten map, is one-loop renormalizable. Meanwhile it suffers from
asymptotic freedom that is not in agreement with the experiment. We show that
QED part of $U_\star(3)\times U_\star(1)$ gauge group as an appropriate gauge
group for the noncommutative QED+QCD, is not only one-loop renormalizable but
also has a $\beta$ function that can be positive, negative and even zero. In
fact the $\beta$ function depends on the mixing parameter $\delta_{13}$ as a
free parameter and it will be equal to its counterpart in the ordinary QED for
$\delta_{13}=0.367\pi$.
