  Let $A$ be a special biserial algebra over an algebraically closed field. We
show that the first Hohchshild cohomology group of $A$ with coefficients in the
bimodule $A$ vanishes if and only if $A$ is representation finite and simply
connected (in the sense of Bongartz and Gabriel), if and only if the Euler
characteristic of $Q$ equals the number of indecomposable non uniserial
projective injective $A$-modules (up to isomorphism). Moreover, if this is the
case, then all the higher Hochschild cohomology groups of $A$ vanish.
