Background: Baum-Welch training is an expectation-maximisation algorithm for training the emission and transition probabilities of hidden Markov models in a fully automated way. Methods and results: We introduce a linear space algorithm for Baum-Welch training. For a hidden Markov model with M states, T free transition and E free emission parameters, and an input sequence of length L, our new algorithm requires O(M) memory and O(L M T_max (T + E)) time for one Baum-Welch iteration, where T_max is the maximum number of states that any state is connected to. The most memory efficient algorithm until now was the checkpointing algorithm with O(log(L) M) memory and O(log(L) L M T_max) time requirement. Our novel algorithm thus renders the memory requirement completely independent of the length of the training sequences. More generally, for an n-hidden Markov model and n input sequences of length L, the memory requirement of O(log(L) L^(n-1) M) is reduced to O(L^(n-1) M) memory while the running time is changed from O(log(L) L^n M T_max + L^n (T + E)) to O(L^n M T_max (T + E)). Conclusions: For the large class of hidden Markov models used for example in gene prediction, whose number of states does not scale with the length of the input sequence, our novel algorithm can thus be both faster and more memory-efficient than any of the existing algorithms.