The Type-I and Type-II censoring schemes are the most prominent and commonly used censoring schemes in practice. In this work, a mixture of Type-I and Type- II censoring schemes, named the Type I-Type II mixture censoring scheme, has been introduced. Different censoring schemes have been discussed, along with their benefits and drawbacks. For the proposed censoring scheme, we analyze the data under the assumption that failure times of experimental units follow the Weibull distribution. The computational formulas for the expected number of failures and the expected failure time are provided. Maximum likelihood estimation and Bayesian estimation are used to estimate the model parameters. On the basis of thorough Monte Carlo simulations, the investigated inferential approaches are evaluated. Finally, a numerical example is provided to demonstrate the method of inference discussed here.