Suppose components have lifetimes which follow a Weibull distribution with both scale and shape parameters unknown. Given the first r failure times of a sample of n components it is required to make predictions on the failure times of the remaining components in the sample. An outline is given of the general Bayesian approach for this problem and that which is appropriate when there is no prior information available. It is shown that prediction bounds on remaining failure times. can be obtained numerically but involve considerable computational effort. The latter can be reduced substantially by restricting the shape parameter to a finite number of possible values, and the possibility of using this method as an approximation to a full Bayesian analysis is discussed. A numerical example is used to illustrate the procedures.