A strongly aperiodic subshift of finite type (SA SFT) on a finitely generated group G is a certain way of labeling the group elements, using only finitely many labels and local rules, so that the labeling can be used to distinguish between group elements. Admitting an SA SFT has consequences: G must be one-ended and have a decidable word problem. One is led naturally to the converse and asks if every one-ended group with decidable word problem admits an SA SFT. Towards this, we prove that every one-ended hyperbolic group admits an SA SFT, vastly generalizing a result of Cohen and Goodman-Strauss with whom this work is joint.