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Abstract: Isolated quantum many-body systems with integrable dynamics generically do not thermalize starting from generic initial states when taken far from equilibrium. As one perturbs such systems away from the integrable point, thermalization sets in, but the nature of the crossover from integrable to thermalizing behavior is an unresolved and actively discussed question. We explore this question by studying the dynamics of the momentum distribution function in a dipolar quantum Newton's cradle consisting of highly magnetic dysprosium atoms. This is accomplished by creating the first ultracold one-dimensional Bose gas with strong magnetic dipole-dipole interactions. These interactions provide tunability of both the strength of the integrability-breaking perturbation and the nature of the near-integrable dynamics. We provide the first experimental evidence that thermalization close to a strongly interacting integrable point occurs in two steps: prethermalization followed by near-exponential thermalization. Moreover, the measured thermalization rate is consistent with a parameter-free theoretical estimate, based on identifying the types of collisions that dominate thermalization. By providing tunability between regimes of integrable and nonintegrable, chaotic dynamics, our work sheds light both on the mechanisms by which isolated quantum many-body systems thermalize, and on the temporal structure of the onset of thermalization.