We present observations of the formation of plate-like snow crystals that provide evidence for an edge-enhancing crystal growth instability. This instability arises when the condensation coefficient describing the growth of an ice prism facet increases as the width of the facet becomes narrower. Coupled with the effects of particle diffusion, this phenomenon causes thin plate-like crystals to develop from thicker prisms, sharpening the edges of the plates to micron or sub-micron dimensions as they grow. This sharpening effect is largely responsible for the formation of thin plate-like ice crystals from water vapor near -15 C, which is a dominant feature in the snow crystal morphology diagram. Other faceted crystalline materials may exhibit similar morphological growth instabilities that promote the diffusion-limited growth of plate-like or needle-like structures.