  The proper spin-foam vertex amplitude is obtained from the EPRL vertex by
projecting out all but a single gravitational sector, in order to achieve
correct semi-classical behavior. In this paper we calculate the gravitational
two-point function predicted by the proper spin-foam vertex to lowest order in
the vertex expansion. We find the same answer as in the EPRL case in the
`continuum spectrum' limit, so that the theory is consistent with the
predictions of linearized gravity in the regime of small curvature. The method
for calculating the two-point function is similar to that used in prior works:
we cast it in terms of an action integral and to use stationary phase methods.
Thus, the calculation of the Hessian matrix plays a key role. Once the Hessian
is calculated, it is used not only to calculate the two-point function, but
also to calculate the coefficient appearing in the semi-classical limit of the
proper vertex amplitude itself. This coefficient is the effective discrete
"measure factor" encoded in the spin-foam model. Through a non-trivial
cancellation of different factors, we find that this coefficient is the same as
the coefficient in front of the term in the asymptotics of the EPRL vertex
corresponding to the selected gravitational sector.
