  We propose a novel artificial compression, reduced order model (AC-ROM) for
the numerical simulation of viscous incompressible fluid flows. The new AC-ROM
provides approximations not only for velocity, but also for pressure, which is
needed to calculate forces on bodies in the flow and to connect the simulation
parameters with pressure data. The new AC-ROM does not require that the
velocity-pressure ROM spaces satisfy the inf-sup (Ladyzhenskaya-Babuska-Brezzi)
condition and its basis functions are constructed from data that are not
required to be weakly-divergence free. We prove error estimates for the reduced
basis discretization of the AC-ROM. We also investigate numerically the new
AC-ROM in the simulation of a two-dimensional flow between offset cylinders.
