  We have synthesized single-crystal samples of $\beta$-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 V$^{4+}$O$_{6}$ octahedra that are linked via 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
$\beta$-VOSO$_4$ is very close to an ideal one-dimensional magnetic system.
