We provide a new extension of general relativity (GR) which has the remarkable property of being more constrained than GR plus a cosmological constant, having one less free parameter. This is implemented by allowing the cosmological constant to have a consistent space-time variation, through coding its dynamics in the torsion tensor. We demonstrate this mechanism by adding a "quasitopological" term to the Einstein action, which naturally realizes a dynamical torsion with an automatic satisfaction of the Bianchi identities. Moreover, variation of the action with respect to this dynamical Λ fixes it in terms of other variables, thus providing a scenario with less freedom than general relativity with a cosmological constant. Once matter is introduced, at least in the homogeneous and isotropic reduction, Λ is uniquely determined by the field content of the model. We make an explicit construction using the Palatini formulation of GR and describe the striking properties of this new theory. We also highlight some possible extensions to the theory. A companion paper [S. Alexander, M. Cortês, A. R. Liddle, J. Magueijo, R. Sims, and L. Smolin, following paper, Cosmology of minimal varying Lambda theories, Phys. Rev. D 100, 083507 (2019), 10.1103/PhysRevD.100.083507] explores the Friedmann-Robertson-Walker reduction for cosmology, and future work will study Solar System tests of the theory.