We study the lengths of monotone subsequences for permutations drawn from the Mallows measure. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation $π$ is proportional to $q^{{\rm inv}(π)}$ where $q$ is a positive parameter and ${\rm inv}(π)$ is the number of inversions in $π$. In our main result we show that when $0<q<1$, the limiting distribution of the longest increasing subsequence (LIS) is Gaussian, answering an open question in [Bhatnagar and Peled, PTRF, 2015]. This is in contrast to the case when $q=1$ where the limiting distribution of the LIS when scaled appropriately is the GUE Tracy-Widom distribution. We also obtain a law of large numbers for the length of the longest decreasing subsequence (LDS) and identify the precise constant in the order of the expectation, answering a further open question in [Bhatnagar and Peled, PTRF, 2015].