This paper studies the problem of testing the null assumption of no-change in the mean of chronologically ordered independent observations on a random variable $X$ {\it versus} the at most one change in the mean alternative hypothesis. The approach taken is via a Darling-Erdős type self-normalized maximal deviation between sample means before and sample means after possible times of a change in the expected values of the observations of a random sample. Asymptotically, the thus formulated maximal deviations are shown to have a standard Gumbel distribution under the null assumption of no change in the mean. A first such result is proved under the condition that $EX^2\log\log (|X|+1)<\infty$, while in the case of a second one, $X$ is assumed to be in a specific class of the domain of attraction of the normal law, possibly with infinite variance.