  S. Smirnov proved recently that the Hausdorff dimension of any K-quasicircle
is at most 1+k^2, where k=(K-1)/(K+1). In this paper we show that if $\Gamma$
is such a quasicircle, then $H^{1+k^2}(B(x,r)\cap \Gamma)\leq C(k) r^{1+k^2}$
for all x in \C and r>0, where H^s stands for the s-Haudorff measure. On a
related note we derive a sharp weak-integrability of the derivative of the
Riemann map of a quasidisk.
