  Classical Sweet-Parker models of reconnection predict that reconnection rates
depend inversely on the resistivity, usually parameterized using the
dimensionless Lundquist number ($\Lund$). We describe magnetohydrodynamic (MHD)
simulations using a static, nested grid that show the development of a
three-dimensional instability in the plane of a current sheet between reversing
field lines without a guide field. The instability leads to rapid reconnection
of magnetic field lines at a rate independent of $\Lund$ over at least the
range $3.2\times 10^3 \lesssim \Lund \lesssim 3.2 \times 10^5$ resolved by the
simulations. We find that this instability occurs even for cases with $\Lund
\lesssim 10^4$ that in our models appear stable to the recently described,
two-dimensional, plasmoid instability. Our results suggest that
three-dimensional, MHD processes alone produce fast (resistivity independent)
reconnection without recourse to kinetic effects or external turbulence. The
unstable reconnection layers provide a self-consistent environment in which the
extensively studied turbulent reconnection process can occur.
