Abstract. We study the controllability to trajectories, under positivity constraints on the control or the state, of a one-dimensional heat equation involving the fractional Laplace operator (−∂2x)s (with 0<s<1) on the interval (−1,1). Our control function is localized in an open set O in the exterior of (−1,1), that is, O⊂(R∖(−1,1)). We show that there exists a minimal (strictly positive) time Tmin such that the fractional heat dynamics can be controlled from any initial datum in L2(−1,1) to a positive trajectory through the action of an exterior positive control, if and only if 12<s<1. In addition, we prove that at this minimal controllability time, the constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. Finally, we provide several numerical illustrations that confirm our theoretical results.