  Newton's gravitational constant G, which determines the strength of
gravitational interactions both in Newton's theory and in Einstein's General
Relativity, is the least well known of all the fundamental constants. Given its
importance, and with recent disparities between experimental measurements, a
new approach is suggested. It is based on a purely gravitational oscillator
without any non-gravitational restoring forces. The suggested technique is
based on the oscillation period of a test mass oscillating through a hole bored
in a solid sphere in free space, or, equivalently, in orbit. The period of
oscillation depends only on the density of the solid sphere, providing a method
with systematic errors different from terrestrial approaches to the
determination of G. Deviations from standard Newtonian gravity and the equality
of gravitational and inertial mass can also be searched for.
