Let θ + X1, θ + X2... be a random sample and Xi's have a common distribution function F, where F has an absolutely continuous densityfthat is symmetric about origin. For a given confidence coefficient, a fully sequential procedure is proposed for constructing a fixed-width confidence interval for the location parameter θ. The result is based on Stone's (1975) adaptive maximum likelihood estimator. under mild conditions the procedure is shown to be asymptotically consistent and fully efficient.