In this work, we study nonsingular bounce cosmology in the context of the Lagrange multiplier generalized F(R) gravity theory of gravity. We specify our study by using a specific variant form of the well-known matter bounce cosmology, with scale factor a(t)=(a0t2+1)n, and we demonstrate that for n<1/2, the primordial curvature perturbations are generated deeply in the contraction era. Particularly, we show explicitly that the perturbation modes exit the horizon at a large negative time during the contraction era, which in turn makes the “low-curvature” regime, the era for which the calculations of observational indices related to the primordial power spectrum can be considered reliable. Using the reconstruction techniques for the Lagrange multiplier F(R) gravity, we construct the form of effective F(R) gravity that can realize such a cosmological evolution, and we determine the power spectrum of the primordial curvature perturbations. Accordingly, we calculate the spectral index of the primordial curvature perturbations and the tensor-to-scalar ratio, and we confront these with the latest observational data. We also address the issue of stability of the primordial metric perturbations, and to this end, we determine the form of F(R) which realizes the nonsingular cosmology for the whole range of cosmic time −∞<t<∞, by solving the Friedmann equations without the “low-curvature” approximation. This study is performed numerically though, due to the high complexity of the resulting differential equations. By using this numerical solution, we show that the stability is achieved for the same range of values of the free parameters that guarantee the phenomenological viability of the model. We also investigate the energy conditions in the present context. The phenomenology of the no-singular bounce is also studied in the context of a standard F(R) gravity. We find that the results obtained in the Lagrange multiplier F(R) gravity model have differences in comparison to the standard F(R) gravity model, where the observable indices are not simultaneously compatible with the latest Planck results, and also the standard F(R) gravity model is plagued with instabilities of the perturbation. These facts clearly justify the importance of the Lagrange multiplier field in making the observational indices compatible with the Planck data and also in removing the instabilities of the metric perturbations. Thereby, the bounce with the aforementioned scale factor is adequately described by the Lagrange multiplier F(R) gravity, in comparison to the standard F(R) model.