We exhibit an empirical Bayes test $\delta_n^*$ for the normal mean
testing problem using a linear error loss. Under the condition that the
critical point of a Bayes test is within some known compact interval,
$\delta_n^*$ is shown to be asymptotically optimal and its associated regret
Bayes risk converges to zero at a rate $O(n^{-1}(\ln n)^{1.5})$, where $n$ is
the number of past experiences available when the current component decision
problem is considered. Under the same condition this rate is faster than the
optimal rate of convergence claimed by Karunamuni.