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The location, shape, and scale parameters of the Weibull distribution are estimated from Type I progressively censored samples by the method of maximum likelihood. Nonlinear logarithmic likelihood estimating equations are derived, and the approximate asymptotic variance-covariance matrix for the maximum likelihood parameter estimates is given. The iterative procedure to solve the likelihood equations is a stable and rapidly convergent constrained modified quasilinearization algorithm which is applicable to the general case in which all three parameters are unknown. The numerical results indicate that, in terms of the number of iterations required for convergence and in the accuracy of the solution, the proposed algorithm is a very effective technique for solving systems of logarithmic likelihood equations for which all iterative approximations to the solution vector must satisfy certain intrinsic constraints on the parameters. A FORTRAN IV program implementing the maximum likelihood estimation procedure is included.