As it was emphasized in the first part of this paper, the problem of relaxation spectrum determination is the practical problem of reconstructing the solution of Fredholm integral equation of the first kind of convolution type from time-measured data. This problem is known to be severely ill-posed. This means that in particular small changes in measured relaxation modulus can lead to arbitrarily large changes in the relaxation spectrum. In remedy, some reduction of the admissible solutions set or respective regularization of the original problem can be used. In this paper both the techniques are used simultaneously. A finite-dimensional approximation of the spectrum by the linear combination of orthogonal functions is combined with Tikhonov regularization. A resulting simple identification scheme is outlined. The effectiveness of the method is demonstrated through the computation of the relaxation spectrum of a sugar beet root sample in the state of uniaxial strain. A numerical example of determining the Gaussian relaxation spectrum is also enclosed.