  We present molecular-line observations of 94 dark cloud cores identified in
the Pipe nebula through near-IR extinction mapping. Using the Arizona Radio
Observatory 12m telescope, we obtained spectra of these cores in the J=1-0
transition of C18O. We use the measured core parameters, i.e., antenna
temperature, linewidth, radial velocity, radius and mass, to explore the
internal kinematics of these cores as well as their radial motions through the
larger molecular cloud. We find that the vast majority of the dark extinction
cores are true cloud cores rather than the superposition of unrelated
filaments. While we identify no significant correlations between the core's
internal gas motions and the cores' other physical parameters, we identify
spatially correlated radial velocity variations that outline two main kinematic
components of the cloud. The largest is a 15pc long filament that is
surprisingly narrow both in spatial dimensions and in radial velocity.
Beginning in the Stem of the Pipe, this filament displays uniformly small C18O
linewidths (dv~0.4kms-1) as well as core to core motions only slightly in
excess of the gas sound speed. The second component outlines what appears to be
part of a large (2pc; 1000 solar mass) ring-like structure. Cores associated
with this component display both larger linewidths and core to core motions
than in the main cloud. The Pipe Molecular Ring may represent a primordial
structure related to the formation of this cloud.
