Let X be a random variable from a normal distribution with unknown mean q and known variance s2. In many practical situations, q is known in advance to lie in an interval, say [- m, m], for some m>0. As the usual estimator of q, i.e., X under the Linex Loss Function is inadmissible, finding some competitors for X becomes worthwhile. The only study in the literature considered the problem of minimax estimation of q In this paper, by constructing a dominating class of estimators, we show that the maximum likelihood estimator is inadmissible. Then, as a competitor, the Bayes estimator associated with a uniform prior on the interval [- m, m] is proposed. Finally, considering risk performance as a comparison criterion, the estimators are compared and depending on the values taken by q in the interval [- m, m], the appropriate estimator is suggested.