  We introduce a simple protocol for adaptive quantum state tomography, which
reduces the worst-case infidelity between the estimate and the true state from
$O(N^{-1/2})$ to $O(N^{-1})$. It uses a single adaptation step and just one
extra measurement setting. In a linear optical qubit experiment, we demonstrate
a full order of magnitude reduction in infidelity (from $0.1%$ to $0.01%$) for
a modest number of samples ($N=3\times10^4$).
