We prove the one-dimensional almost sure invariance principle with essentially optimal rates for slowly (polynomially) mixing deterministic dynamical systems, such as Pomeau-Manneville intermittent maps, with Hölder continuous observables. Our rates have form o(n γ L(n)), where L(n) is a slowly varying function and γ is determined by the speed of mixing. We strongly improve previous results where the best available rates did not exceed O(n 1/4). To break the O(n 1/4) barrier, we represent the dynamics as a Young-tower-like Markov chain and adapt the methods of Berkes-Liu-Wu and Cuny-Dedecker-Merlevède on the Komlós-Major-Tusnády approximation for dependent processes.