Current cone jet algorithms, widely used at hadron colliders, take event particles as seeds in an iterative search for stable cones. A longstanding infrared (IR) unsafety issue in such algorithms is often assumed to be solvable by adding extramidpoint'seeds, but actually is just postponed to one order higher in the coupling. A proper solution is to switch to an exact seedless cone algorithm, one that provably identifies all stable cones. The only existing approach takes N2 N time to find jets among Nparticles, making it unusable at hadron level. This can be reduced to N 2 ln N time, leading to code (SISCone) whose speed is similar to that of public midpoint implementations. Monte Carlo tests provide a strong cross-check of an analytical proof of the IR safety of the new algorithm, and the absence of anyR sep'issue implies a good practical correspondence between parton and hadron levels. Relative to a midpoint cone, the use of an IR safe seedless algorithm leads to modest changes for inclusive jet spectra, mostly through reduced sensitivity to the underlying event, and significant changes for some multi-jet observables.