A. We prove that the fractional heat equations posed on the whole Euclidean space ℝ and associated with the operators (−∆) ∕2 are exactly null-controllable from control supports which are sufficiently "exponentially thick", when 0 < ≤ 1. This is a first positive null-controllability result in this weak dissipation regime. When 0 < < 1, we also give a necessary condition of the same form. This notion of exponential thickness is stronger than the geometric notion of thickness known to be a necessary and sufficient condition to obtain positive null-controllability results in the strong dissipation regime > 1. Inspired by the construction of the Smith-Volterra-Cantor sets, we also provide examples of non-trivial exponentially thick control supports.