  We show by explicit construction that for every solution of the
incompressible Navier-Stokes equation in $p+1$ dimensions, there is a uniquely
associated "dual" solution of the vacuum Einstein equations in $p+2$
dimensions. The dual geometry has an intrinsically flat timelike boundary
segment $\Sigma_c$ whose extrinsic curvature is given by the stress tensor of
the Navier-Stokes fluid. We consider a "near-horizon" limit in which $\Sigma_c$
becomes highly accelerated. The near-horizon expansion in gravity is shown to
be mathematically equivalent to the hydrodynamic expansion in fluid dynamics,
and the Einstein equation reduces to the incompressible Navier-Stokes equation.
For $p=2$, we show that the full dual geometry is algebraically special Petrov
type II. The construction is a mathematically precise realization of
suggestions of a holographic duality relating fluids and horizons which began
with the membrane paradigm in the 70's and resurfaced recently in studies of
the AdS/CFT correspondence.
