Abstract: We investigate in detail the interaction between the spin-(1/2) field endowed with mass dimension one and the graviton. We obtain an interaction vertex that combines the characteristics of scalar-graviton and Dirac's fermion-graviton vertices, due to the scalar-dynamic attribute and the fermionic structure of the mass-dimension-one field. It is shown that this vertex obeys the Ward-Takahashi identity, ensuring the gauge invariance for the interaction. In the contribution of the mass-dimension-one fermion to the graviton propagator at one-loop level, we found the conditions for the cancellation of the tadpole term by a cosmological counterterm. We calculate the scattering process for arbitrary momentum. For low energies, the result reveals that only the scalar sector present in the vertex contributes to the gravitational potential. Finally, we evaluate the non-relativistic limit of the gravitational interaction and obtain an attractive Newtonian potential, as required for a dark-matter candidate.