We consider a double sampling plan in which the first sample can be referred to as a pilot sample. A pilot study is often done prior to a more thorough study to determine whether further sampling should be conducted. If the second sample is taken, then the data from these two samples can be used for inference purposes. This paper studies the confidence interval of the Weibull shape (extreme-value scale) parameter using this double sampling procedure. The confidence interval obtained from the two samples constitutes an exact conditional confidence interval (CCI) for the parameter. The conditional coverage probability of a usual confidence interval (UCI) is obtained by computing the coverage probability under the conditional probability density function. It is shown that the conditional coverage probability of the UCI is not uniformly greater than or less than the nominal level.