  We investigate in detail the interaction between the spin-${1/2}$ fields
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 this field. It is shown that the vertex obtained obeys
the Ward-Takahashi identity, ensuring the gauge invariance for this
interaction. In the contribution of the mass dimension one fermion to the
graviton propagator at one-loop, we found the conditions for the cancellation
of the tadpole term by a cosmological counter-term. 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.
