Phase separation is a central concept of materials physics [1-3] and has recently emerged as an important route to compartmentalization within living cells [4-6]. Biological phase separation features activity , complex compositions , and elasticity , which reveal important gaps in our understanding of this universal physical phenomenon. Here, we explore the impact of elasticity on phase separation in synthetic polymer networks. We show that compressive stresses in a polymer network can suppress phase separation of the solvent that swells it, stabilizing mixtures well beyond the liquid-liquid phase separation boundary. Network stresses also drive a new form of ripening, driven by transport of solute down stiffness gradients. This elastic ripening can be much faster than conventional surface tension driven Ostwald ripening.