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Let $(M,g,\si)$ be a compact spin manifold of dimension $n \geq 2$. Let$\lambda_1^+(\tilde{g})$ be the smallest positive eigenvalue of the Diracoperator in the metric $\tilde{g} \in [g]$ conformal to $g$. We then define$\lamin(M,[g],\si) = \inf_{\tilde{g} \in [g]} \lambda_1^+(\tilde{g})\Vol(M,\tilde{g})^{1/n} $. We show that $0< \lamin(M,[g],\si) \leq\lamin(\mS^n)$. %=\frac{n}{2} \om_n^{{1 \over n}}$ . We find sufficientconditions for which we obtain strict inequality $\lamin(M,[g],\si) <\lamin(\mS^n)$. This strict inequality has applications to conformal spingeometry.    Soit $(M,g,\si)$ une vari\'et\'e spinorielle compacte de dimension $n \geq2$. %Si $\tilde{g} \in [g]$ est une m\'etrique conforme \`a $g$, On note$\lambda_1^+(\tilde{g})$ la plus petite valeur propre $>0$ de l'op\'erateur deDirac dans la m\'etrique $\tilde{g} \in [g]$ conforme \`a $g$. On d\'efinit$\lamin(M,[g],\si) = \inf_{\tilde{g} \in [g]} \lambda_1^+(\tilde{g})  \Vol(M,\tilde{g})^{1/n} $. On montre que $0< \lamin(M,[g],\si) \leq\lamin(\mS^n)$. %= \frac{n}{2} \om_n^{{1 \over n}}$ On trouve des conditionssuffisantes pour lesquelles on obtient l'in\'egalit\'e stricte$\lamin(M,[g],\si) < \lamin(\mS^n)$. Cette in\'egalit\'e stricte a desapplications en g\'eom\'etrie spinorielle conforme.

Un probl\`eme de type Yamabe sur les vari\'et\'es compactes spinorielles compactes