In the quest for superconductors with high transition temperatures (T\(_\mathrm{c}\)s), one emerging motif is that unconventional superconductivity is enhanced by fluctuations of a broken-symmetry phase near a quantum-critical point. While recent experiments have suggested the existence of the requisite broken symmetry phase in the high-T\(_\mathrm{c}\) cuprates, the signature of quantum-critical fluctuations in the electronic structure has thus far remained elusive, leaving their importance for high-T\(_\mathrm{c}\) superconductivity in question. We use magnetic fields exceeding 90 tesla to access the underlying metallic state of the cuprate YBa2Cu3O6+\(_\delta\) over an unprecedented range of doping, and magnetic quantum oscillations reveal a strong enhancement in the quasiparticle effective mass toward optimal doping. This mass enhancement is a characteristic signature of quantum criticality, and identifies a quantum-critical point at p\(_{crit}\) \(\approx\) 0.18. This point also represents the juncture of the vanishing pseudogap energy scale and the disappearance of Kerr rotation, the negative Hall coefficient, and the recently observed charge order, suggesting a mechanism of high-T\(_\mathrm{c}\) that is strongest when these definitive experimental signatures of the underdoped cuprates converge at a quantum critical point.