  Large-scale, dust-trapping vortices may account for observations of
asymmetric protoplanetary discs. Disc vortices are also potential sites for
accelerated planetesimal formation by concentrating dust grains. However, in 3D
discs vortices are subject to destructive `elliptic instabilities', which
reduces their viability as dust traps. The survival of vortices in 3D accretion
discs is thus an important issue to address. In this work, we perform shearing
box simulations to show that disc self-gravity enhances the survival of 3D
vortices, even when self-gravity is weak in the classic sense (e.g. with a
Toomre $Q\simeq5$). We find a 3D, self-gravitating vortex can grow on secular
timescales in spite of the elliptic instability. The vortex aspect-ratio
decreases as it strengthens, which feeds the elliptic instability. The result
is a 3D vortex with a turbulent core that persists for $\sim 10^{3}$ orbits. We
find when gravitational and hydrodynamic stresses become comparable, the vortex
may undergo episodic bursts, which we interpret as interaction between elliptic
and gravitational instabilities. We estimate the distribution of dust particles
in self-gravitating, turbulent vortices. Our results suggest large-scale
vortices in protoplanetary discs are more easily observed at large radii.
