In sampling from a finite population, the nonexistence of a uniformly minimum variance unbiased estimator for the mean $\mu$ has been demonstrated by Godambe [3], and the inadmissibility of the sample mean as an estimator for $\mu$, when sampling is with replacement and equal probabilities, has been proved by Des Raj and Khamis [2] and by Basu [1]. In this paper, the problem of unbiased linear estimation of $\mu$ with minimum variance is considered for a very general scheme of sampling. An admissible estimator is obtained, together with a complete class of estimators. It is shown further that, for a somewhat restricted sampling scheme, amongst estimators with variance proportional to $\sigma^2$, there does exist a best estimator which, in the case of sampling with replacement and equal probabilities, is the same as that considered in [1] and [2].