When a globally supersymmetric theory is scale invariant, it must possess a
Virial supercurrent supermultiplet. The multiplet structure is analogous to the
R-current supermultiplet in globally R-symmetric theories but we put extra
"$i$"s in various formulae. We construct a novel type of supergravity from
gauging the Virial supercurrent supermultiplet in $d=1+3$ dimensions. We give
the full non-linear superspace action with the help of a covariantly linear
unitary superconformal compensator. The resulting supergravity is peculiar: (1)
no Einstein-like second order kinetic term is allowed without matter (2) there
exists a dynamical non-geometrical connection (3) the metric is unimodular in
the Wess-Zumino gauge and can be coupled only to scale invariant matter.
Examples of matter couplings and higher derivative kinetic terms are studied
from the superconformal approach. Our work completes the classification of the
irreducible $\mathcal{N}=1$ supergravities in $d=1+3$ dimensions.