In this paper a novel constrained-stability least-mean-squares (LMS) algorithm for filtering speech sounds is proposed in the adaptive noise cancellation (ANC) problem. It is based on the minimization of the squared Euclidean norm of the weight vector change under a stability constraint over the a posteriori estimation errors. To this purpose, the Lagrangian methodology has been used in order to propose a nonlinear adaptation in terms of the product of differential input and error. Convergence analysis is also studied in terms of the evolution of the natural modes to the optimal Wiener-Hopf solution so that the stability performance depends exclusively on the adaptation parameter mu and the eigenvalues of the difference matrix DeltaR(1). The algorithm shows superior performance over the referenced algorithms in the ANC problem of speech discontinuous transmission systems, which are characterized by rapid transitions of the desired signal. The experimental analysis carried out on the AURORA 3 speech databases provides an extensive performance evaluation together with an exhaustive comparison to the standard LMS algorithms, i.e., the normalized LMS (NLMS), and other recently reported LMS algorithms such as the modified NLMS, the error nonlinearity LMS, or the normalized data nonlinearity LMS adaptation.