Homeostasis is a recurring theme in biology that ensures that regulated variables robustly-and in some systems, completely-adapt to environmental perturbations. This robust perfect adaptation feature is achieved in natural circuits by using integral control, a negative feedback strategy that performs mathematical integration to achieve structurally robust regulation(1,2). Despite its benefits, the synthetic realization of integral feedback in living cells has remained elusive owing to the complexity of the required biological computations. Here we prove mathematically that there is a single fundamental biomolecular controller topology(3) that realizes integral feedback and achieves robust perfect adaptation in arbitrary intracellular networks with noisy dynamics. This adaptation property is guaranteed both for the population-average and for the time-average of single cells. On the basis of this concept, we genetically engineer a synthetic integral feedback controller in living cells(4) and demonstrate its tunability and adaptation properties. A growth-rate control application in Escherichia coli shows the intrinsic capacity of our integral controller to deliver robustness and highlights its potential use as a versatile controller for regulation of biological variables in uncertain networks. Our results provide conceptual and practical tools in the area of cybergenetics(3,5), for engineering synthetic controllers that steer the dynamics of living systems(3-9).