  We continue the study of the non-metric theory of gravity introduced in
hep-th/0611182 and gr-qc/0703002 and obtain its general spherically symmetric
vacuum solution. It respects the analog of the Birkhoff theorem, i.e., the
vacuum spherically symmetric solution is necessarily static. As in general
relativity, the spherically symmetric solution is seen to describe a black
hole. The exterior geometry is essentially the same as in the Schwarzschild
case, with power-law corrections to the Newtonian potential. The behavior
inside the black-hole region is different from the Schwarzschild case in that
the usual spacetime singularity gets replaced by a singular surface of a new
type, where all basic fields of the theory remain finite but metric ceases to
exist. The theory does not admit arbitrarily small black holes: for small
objects, the curvature on the would-be horizon is so strong that non-metric
modifications prevent the horizon from being formed. The theory allows for
modifications of gravity of very interesting nature. We discuss three physical
effects, namely, (i) correction to Newton's law in the neighborhood of the
source, (ii) renormalization of effective gravitational and cosmological
constants at large distances from the source, and (iii) additional redshift
factor between spatial regions of different curvature. The first two effects
can be responsible, respectively, for the observed anomaly in the acceleration
of the Pioneer spacecraft and for the alleged missing mass in spiral galaxies
and other astrophysical objects. The third effect can be used to propose a
non-cosmological explanation of high redshifts of quasars and gamma-ray bursts.
