  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 extra `midpoint' 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 N 2^N time to find jets among N particles, 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 any
'R_{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.
