  In this paper, we give the H\"ormander's $L^2$ theorem for Dirac operator
over an open subset $\Omega\in\R^{n+1}$ with Clifford algebra. Some sufficient
condition on the existence of the weak solutions for Dirac operator has been
found in the sense of Clifford analysis. In particular, if $\Omega$ is bounded,
then we prove that for any $f$ in $L^2$ space with value in Clifford algebra,
there exists a weak solution of Dirac operator such that $$\bar{D}u=f$$ with
$u$ in the $L^2$ space as well. The method is based on H\"ormander's $L^2$
existence theorem in complex analysis and the $L^2$ weighted space is utilised.
