Star formation in the Galactic disc is primarily controlled by gravity, turbulence, and magnetic fields. It is not clear that this also applies to star formation near the Galactic Centre. Here we determine the turbulence and star formation in the CMZ cloud G0.253+0.016. Using maps of 3 mm dust emission and HNCO intensity-weighted velocity obtained with ALMA, we measure the volume-density variance σρ /ρ 0=1.3±0.5 and turbulent Mach number
= 11±3. Combining these with turbulence simulations to constrain the plasma β = 0.34±0.35, we reconstruct the turbulence driving parameter b=0.22±0.12 in G0.253+0.016. This low value of b indicates solenoidal (divergence-free) driving of the turbulence in G0.253+0.016. By contrast, typical clouds in the Milky Way disc and spiral arms have a significant compressive (curl-free) driving component (b > 0.4). We speculate that shear causes the solenoidal driving in G0.253+0.016 and show that this may reduce the star formation rate by a factor of 7 compared to nearby clouds.