  We use recent information on geodetic precession of the binary pulsar
B1913+16 along with measurements of its orbital parameters and proper motion to
derive new constraints on the immediate progenitor of this double neutron star
system. As part of our analysis we model the motion of the binary in the Galaxy
after the second supernova explosion, and we derive constraints on the unknown
radial velocity. We also obtain limits on the magnitude and direction of the
kick velocity imparted to the pulsar companion during the second supernova
explosion. We consider the complete set of possible cases, depending on the
kinematic age of the system and the 180 degree ambiguity in the pulsar spin
orientation. Most interestingly, we find that the natal kick must have been
directed almost perpendicular to the spin axis of the neutron star progenitor,
independent of the specific pre-supernova configuration. Such a tight
constraint on the kick direction has important implications for the physical
mechanism responsible for the kick.
