We consider the problem of sample degeneracy in Approximate Bayesian computation (ABC). Sample degeneracy arises when proposed parameters values, once given as an input to the generative model, rarely lead to simulations resembling the observed data, and are therefore discarded. Such ``poor'' parameters proposals do not contribute at all to the representation of the posterior distribution. This leads to a huge number of required simulations and/or a waste of computational resources, as well as to distortions in the computed posterior distribution. To mitigate the sample degeneracy problem, we propose a Large Deviations Weighted Approximate Bayesian Computation algorithm (LDW-ABC), where, via Large Deviations Theory (LDT), strictly positive weights are computed for all parameters proposals, thus avoiding the rejection step altogether. In order to make the integration of LDT into ABC as smooth as possible, we adopt an information theoretic formulation known as the Method of Types, thus restricting our attention to models for discrete random variables. Finally, we evaluate the performances of our methodology through examples on i.i.d data and Markov processes.