The spins and parities of low-lying states in $^{72}\mathrm{Br}$ populated in the $\ensuremath{\beta}$ decay of $^{72}\mathrm{Kr}$ have been studied via conversion electron spectroscopy. The measurements were carried out at ISOLDE using a miniorange spectrometer with Si(Li) and HPGe detectors for electrons and $\ensuremath{\gamma}$ ray detection. Results of the conversion coefficients corresponding to transitions deexciting 12 levels in $^{72}\mathrm{Br}$ are reported. The multipolarities of the transitions are deduced and the spins and parities of the levels involved are discussed. From the multipolarities of the most intense transitions to the ground state, the spin and parity of the $^{72}\mathrm{Br}$ ground state have been definitely established as ${1}^{+}$. The spin of the 101.2-keV isomeric state is determined to be ${3}^{\ensuremath{-}}$. The level scheme is compared with mean-field and shell-model calculations and oblate deformation for the $^{72}\mathrm{Br}$ ground state is deduced. No $E0$ transitions have been found in $^{72}\mathrm{Br}$. $E0$ transitions in the neighboring isobaric nuclei, $^{72}\mathrm{Se}$ and $^{72}\mathrm{Ge}$, have also been studied.