  The {\it exact} formulation for the effect of the Brans-Dicke scalar field on
the gravitational corrections to the Sagnac delay in the Jordan and Einstein
frames is presented for the first time. The results completely agree with the
known PPN factors in the weak field region. The calculations also reveal how
the Brans-Dicke coupling parameter (appears in various correction terms for
different types of source/observer orbits. A first order correction of roughly
2.83 x 10^{-1} fringe shift for visible light is introduced by the
gravity-scalar field combination for Earth bound equatorial orbits. It is also
demonstrated that the final predictions in the two frames do not differ. The
effect of the scalar field on the geodetic and Lense-Thirring precession of a
spherical gyroscope in circular polar orbit around the Earth is also computed
with an eye towards the Stanford Gravity Probe-B experiment currently in
progress. The feasibility of optical and matter-wave interferometric
measurements is discussed briefly.
