In this Letter, we report on the creation of hard x-ray beams carrying orbital angular momentum of topological charge -ℏ and -3ℏ at a photon energy of 8.2 keV via spiral phase plates made out of fused silica by ultrashort-pulsed laser ablation. The phase plates feature a smooth phase ramp with a 0.5 μm nominal step height and a surface roughness of 0.5 μm. The measured vortex beams show submicrometer-sized donut rings and agree well with numerical modeling. Fused silica phase plates are potentially suited to manipulate the electromagnetic field in highly intense x-ray beams at x-ray free-electron laser sources.