We propose AdaTS, a Thompson sampling algorithm that adapts sequentially to bandit tasks that it interacts with. The key idea in AdaTS is to adapt to an unknown task prior distribution by maintaining a distribution over its parameters. When solving a bandit task, that uncertainty is marginalized out and properly accounted for. AdaTS is a fully-Bayesian algorithm that can be implemented efficiently in several classes of bandit problems. We derive upper bounds on its Bayes regret that quantify the loss due to not knowing the task prior, and show that it is small. Our theory is supported by experiments, where AdaTS outperforms prior algorithms and works well even in challenging real-world problems.