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A new Bayesian modelling framework is introduced for piece-wise homogeneous variable-memory Markov chains, along with a collection of effective algorithmic tools for change-point detection and segmentation of discrete time series. Building on the recently introduced Bayesian Context Trees (BCT) framework, the distributions of different segments in a discrete time series are described as variable-memory Markov chains. Inference for the presence and location of change-points is then performed via Markov chain Monte Carlo sampling. The key observation that facilitates effective sampling is that, using one of the BCT algorithms, the prior predictive likelihood of the data can be computed exactly, integrating out all the models and parameters in each segment. This makes it possible to sample directly from the posterior distribution of the number and location of the change-points, leading to accurate estimates and providing a natural quantitative measure of uncertainty in the results. Estimates of the actual model in each segment can also be obtained, at essentially no additional computational cost. Results on both simulated and real-world data indicate that the proposed methodology performs better than or as well as state-of-the-art techniques.