For stability of biorthogonal wavelet bases associated with finite filter banks, two related Lawton matrices must have a simple eigenvalue at one and all remaining eigenvalues of modulus less than one. If the filters are perturbed these eigenvalues must be re-calculated to determine the stability of the new bases — a numerically intensive task. We present a simpler stability criterion. Starting with stable biorthogonal wavelet bases we perturb the associated filters while ensuring that the new Lawton matrices continue to have an eigenvalue at one. We show that stability of the new biorthogonal wavelet bases first breaks down, not just when a second eigenvalue attains a modulus of one, but rather when this second eigenvalue actually equals one. Stability is therefore established by counting eigenvalues at one of finite matrices. The new criterion, in conjunction with the lifting scheme, provides an algorithm for the custom design of stable filter banks.