Using some available information about the unknown variance $sigma$$^2$ of a normal distribution with mean $mu$, a sequential approach is used to estimate $sigma$$^2$. Two cases have been considered regarding the mean $mu$ being known or unknown. The mean square error (MSE) of the new estimators are compared to that of the usual estimator of $sigma$$^2$, namely, the sample variance based on a sample of size equal to the expected sample size. Simulation results indicates that, the new estimator is more efficient than the usual estimator of $sigma$$^2$whenever the actual value of $sigma$$^2$ is not too far from the prior information.