Theories with spontaneously broken conformal symmetry are ubiquitous in models for new physics, with examples including composite Higgs and conformal dark sectors. These models require a mechanism to stabilize the scale of symmetry breaking. Moreover, the stabilization mechanism affects the phenomenology of the dilaton, the pseudo-Goldstone boson of broken scale invariance, as well as the conformal phase transition. I will introduce a new stabilization mechanism in which a relevant operator triggers spontaneous symmetry breaking, in contrast to the usual Goldberger-Wise mechanism involving a near-marginal operator. This "relevant stabilization" has important consequences for the conformal phase transition: we find a weaker first-order phase transition without substantial supercooling. Consequently, the gravitational wave signals from the phase transition are smaller, although still observable at next-generation space-based interferometers.