We have synthesized single-crystal samples of $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{VOSO}}_{4}$ and fully characterized their magnetic properties. Our magnetic susceptibility, high field magnetization, and powder inelastic neutron scattering results are in excellent agreement with theoretical expressions for a one-dimensional spin-1/2 Heisenberg chain with an exchange parameter of $3.83(2)$ meV. Ab initio calculations identify the superexchange pathway, revealing that the spin-chain does not run along the expected crystallographic chain $a$ direction but instead between ${\mathrm{V}}^{4+}{\mathrm{O}}_{6}$ octahedra that are linked via ${\mathrm{SO}}_{4}$ tetrahedra along the $b$ axis. We do not detect any phase transition to a long-range magnetic order within our experimental conditions, indicating $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{VOSO}}_{4}$ is very close to an ideal one-dimensional magnetic system.