Probability estimators developed previously by the authors have been used to obtain unbiased estimates of the Weibull parameters by the linear regression method. Using these unbiased estimators, percentiles of the Weibull distribution have been estimated. Since these percentiles are determined from the estimated parameters, they also have distributions and subsequently are determined for five sample sizes. Analysis has shown that the distributions of these estimated percentiles are neither normal, lognormal, three-parameter Weibull nor three-parameter log-Weibull. A new methodology to estimate the percentile with a specified level of confidence has been introduced. The step-by-step use of the methodology is demonstrated by examples in this paper.