A principal technical result of this paper is that the one-dimensional heat equation with boundary control is exactly null-controllable with control restricted to an arbitrary set epsilon subset of [0, T] of positive measure, A general abstract argument is presented to show that, in contrast to previous results, this implies the bang-bang property ibr time-optimal controls - i.e., such a control can take only extreme values of (the hull of) the constraint set - without imposing any condition regarding the target state.