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In this paper, we propose the first solution to the long-standing problem of designing a globally convergent direct adaptive pole-placement controller for linear, time-invariant discrete-time systems with arbitrary zeros that does not rely on persistency of excitation assumptions. As is well known, the main difficulty of this design is that it involves the estimation of parameters that enter nonlinearly in the regression model. This problem can be overcome introducing an overparameterized representation of the system, which imposes very strict persistency of excitation conditions to prove the parameter convergence. The latter is avoided here using a new version of the dynamic regressor extension and mixing parameter estimator recently proposed in the literature. The main feature of this estimator is that it generates, out of an m-dimensional vector regression, m scalar regression models. This property allows us to estimate only the controller parameters of interest for the adaptive implementation, whose convergence is ensured under assumptions that are strictly weaker than the classical persistency of excitation requirement. Simulation results that illustrate the performance of the proposed scheme are also presented.