  We present the data of the Plateau de Bure Arcsecond Whirlpool Survey (PAWS),
a high spatial and spectral resolution 12CO(1-0) line survey of the inner 10 x
6 kpc of the M51 system, and the first wide-field imaging of molecular gas in a
star-forming spiral galaxy with resolution matched to the typical size of Giant
Molecular Clouds (40 pc). We describe the observation, reduction, and
combination of the Plateau de Bure Interferometer (PdBI) and IRAM-30m "short
spacing" data. The final data cube attains 1.1"-resolution over the 270" x 170"
field of view, with sensitivity to all spatial scales from the combination of
PdBI and IRAM-30m data, and brightness sensitivity of 0.4 K (1 sigma) in each 5
km/s-wide channel map. We find a CO-luminosity of 9.10^8 K km/s pc^2,
corresponding to a molecular gas mass of 4.10^9 Msol for a standard CO-to-H2
conversion factor. Unexpectedly, we find that a large fraction, (50 +/- 10)%,
of this emission arises mostly from spatial scales larger than 36" or about 1.3
kpc. Through a series of tests, we demonstrate that this extended emission does
not result from a processing artifact. We discuss its origin in light of the
stellar component, the 12CO/13CO ratio, and the difference between the
kinematics and structure of the PdBI-only and hybrid synthesis (PdBI +
IRAM-30m) images. The extended emission is consistent with a thick, diffuse
disk of molecular gas with a typical scale height of about 200 pc,
substructured in unresolved filaments which fills about 0.1% of the volume.
