It has been well established that both in bulk at ambient pressure and for films under modest strains, cubic SrCoO$_{3-delta}$ ($delta < 0.2$) is a ferromagnetic metal. Recent theoretical work, however, indicates that a magnetic phase transition to an antiferromagnetic structure could occur under large strain accompanied by a metal-insulator transition. We have observed a strain-induced ferromagnetic to antiferromagnetic phase transition in SrCoO$_{3-delta}$ films grown on DyScO$_3$ substrates, which provide a large tensile epitaxial strain, as compared to ferromagnetic films under lower tensile strain on SrTiO$_3$ substrates. Magnetometry results demonstrate the existence of antiferromagnetic spin correlations and neutron diffraction experiments provide a direct evidence for a G-type antiferromagnetic structure with Neel temperatures between $T_N sim 135,pm,10,K$ and $sim 325,pm,10,K$ depending on the oxygen content of the samples. Therefore, our data experimentally confirm the predicted strain-induced magnetic phase transition to an antiferromagnetic state for SrCoO$_{3-delta}$ thin films under large epitaxial strain.