Using fully kinetic simulations with a Fokker-Planck collision operator, it is demonstrated that Sweet-Parker reconnection layers are unstable to plasmoids (secondary islands) for Lundquist numbers beyond S greater, similar 1000. The instability is increasingly violent at higher Lundquist numbers, both in terms of the number of plasmoids produced and the super-Alfvénic growth rate. A dramatic enhancement in the reconnection rate is observed when the half-thickness of the current sheet between two plasmoids approaches the ion inertial length. During this transition to kinetic scales, the reconnection electric field rapidly exceeds the runaway limit, resulting in the formation of electron-scale current layers that are unstable to the continual formation of new plasmoids.