  In Monte Carlo integration an accurate and reliable determination of the
numerical intregration error is essential. We point out the need for an
independent estimate of the error on this error, for which we present an
unbiased estimator. In contrast to the usual (first-order) error estimator,
this second-order estimator can be shown to be not necessarily positive in an
actual Monte Carlo computation. We propose an alternative and indicate how this
can be computed in linear time without risk of large rounding errors. In
addition, we comment on the relatively very slow convergence of the
second-order error estimate.
