The Standard Model Higgs potential may become unbounded from below at large field values $h\gtrsim h_{\mathrm{top}}\sim {10}^{10}\mathrm{GeV}$, with important cosmological implications. For a potential of this form, the commonly assumed scenario of a nucleated thin-wall bubble driving the transition from the electroweak vacuum to the unstable region does not apply. We present exact analytical solutions for potentials that have the same qualitative form as the Higgs potential. They show that the transition is driven by a thick-wall spherical bubble of true vacuum, with a surface that expands at asymptotically the speed of light. A “crunch” singularity appears in the quasi-anti–de Sitter interior, with the collapsed region also expanding at asymptotically the speed of light. The singularity is surrounded by a region of trapped surfaces whose boundary forms an apparent horizon. An event horizon separates the singularity from the bubble exterior, so that the expansion of the bubble surface is not affected by the collapse of the interior. The solutions provide exact descriptions of the geometry for thick-wall bubbles and are consistent with the analysis of Espinosa et al. [J. High Energy Phys. 09 (2015) 174.] and Strumia-Tetradis [J. High Energy Phys. 09 (2022) 203.] for the Higgs potential.