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A new method of constructing a universal sequence of block codes for coding a class of ergodic sources is given. With this method, a weakly universal sequence of codes is constructed for variable-rate noise. less coding and for fixed- and variable-rate coding with respect to a fidelity criterion. In this way a unified approach to weak universal block source coding is obtained. For the noiseless variable-rate coding and the fixed-rate coding with respect to fidelity criterion, the assumptions made on the alphabets, distortion measures, and class of sources are both necessary and sufficient. For fixed-rate coding with respect to a fidelity criterion, the sample distortion of the universal code sequence converges in norm for each source to the optimum distortion for that source. For both variable-rate noiseless coding and variable-rate coding with respect to a fidelity criterion, the sample rate of the universal code sequence converges in norm for each source to the optimum rate for that source. Using this fact, a universal sequence of codes for fixed-rate noiseless coding is obtained. Some applications to stationary nonergodic sources are also considered. The results of Davisson, Ziv, Neuhoff, Gray, Pursley, and Mackenthun are extended.