We describe a new mechanism of dark matter production. If dark matter particles acquire mass during a first order phase transition, it is energetically unfavorable for them to enter the expanding bubbles. Instead, most of them are reflected and quickly annihilate away. The bubbles eventually merge as the phase transition completes and only the dark matter particles that have entered the bubbles survive to constitute the observed dark matter today. This mechanism can produce dark matter with masses from the TeV scale to above the PeV scale, surpassing the Griest-Kamionkowski bound.