We show that a spin-$5/2$ field can be consistently coupled to gravitation without cosmological constant in five-dimensional spacetimes. The fermionic gauge "hypersymmetry" requires the presence of a finite number of additional fields, including a couple of $U(1)$ fields, a spinorial two-form, the dual of the graviton (of mixed $(2,1)$ Young symmetry) and a spin-$3$ field... The gravitational sector of the action is described by the purely quadratic Gauss-Bonnet term, so that the field equations for the metric are of second order. The local gauge symmetries of the full action principle close without the need of auxiliary fields. The field content corresponds to the components of a connection for an extension of the "hyper-Poincar\'e" algebra, which apart from the Poincar\'e and spin-$3/2$ generators, includes a generator of spin $2$ and a $U(1)$ central extension. It is also shown that this algebra admits an invariant trilinear form, which allows to formulate hypergravity as a gauge theory described by a Chern-Simons action in five dimensions. (read more)