Reliability demonstration tests (RDTs) to ensure a pre-specified reliability target with a stated confidence level are widely used in industry. A new two-stage RDT plan that is more efficient in terms of expected test duration and useful in practice than corresponding single-stage ones is proposed for Weibull and lognormal distributions, respectively. Accepting zero or one failure two-stage plans to minimize the expected test duration at various quality levels including a lifetime target are proposed and analyzed under Type I censoring and sample sizes designated. Numerical examples are studied to illustrate the proposed two-stage RDT plans.