  We prove the analogue of Weyl's law for a noncommutative Riemannian manifold,
namely the noncommutative two torus $\mathbb{T}_\theta^2$ equipped with a
general translation invariant conformal structure and a Weyl conformal factor.
This is achieved by studying the asymptotic distribution of the eigenvalues of
the perturbed Laplacian on $\mathbb{T}_\theta^2$. We also prove the analogue of
Connes' trace theorem by showing that the Dixmier trace and a noncommutative
residue coincide on pseudodifferential operators of order -2 on
$\mathbb{T}_\theta^2$.
