  We present results and applications of high precision timing measurements of
the millisecond pulsar J1012+5307. Combining our radio observations with
results based on optical observations, we derive complete 3-D velocity
information for this system. Correcting for Doppler effects, we derive the
intrinsic spin parameters of this pulsar and a characteristic age of 8.6 +/-
1.9 Gyr. Our upper limit for the orbital eccentricity of only 8 * 10^-7 (68%
C.L.) is the smallest ever measured for a binary system. We demonstrate that
this makes the pulsar an ideal laboratory to test certain aspects of
alternative theories of gravitation. Our precise measurements suggest
deviations from a simple pulsar spin-down timing model, which are consistent
with timing noise and the extrapolation of the known behaviour of slowly
rotating pulsars.
