  Let $(M,g,\si)$ be a compact Riemannian spin manifold of dimension $\geq 2$.
For any metric $\tilde g$ conformal to $g$, we denote by $\tilde\lambda$ the
first positive eigenvalue of the Dirac operator on $(M,\tilde g,\si)$. We show
that $$\inf_{\tilde{g} \in [g]} \tilde\lambda \Vol(M,\tilde g)^{1/n} \leq (n/2)
\Vol(S^n)^{1/n}.$$ This inequality is a spinorial analogue of Aubin's
inequality, an important inequality in the solution of the Yamabe problem. The
inequality is already known in the case $n \geq 3$ and in the case $n = 2$,
$\ker D=\{0\}$. Our proof also works in the remaining case $n=2$, $\ker D\neq
\{0\}$. With the same method we also prove that any conformal class on a
Riemann surface contains a metric with $2\tilde\lambda^2\leq \tilde\mu$, where
$\tilde\mu$ denotes the first positive eigenvalue of the Laplace operator.
