In this letter, we report a new mass for $^{11}$Li using the trapping experiment TITAN at TRIUMFs ISAC facility. This is by far the shortest-lived nuclide, $t_{1/2} = 8.8 rm{ms}$, for which a mass measurement has ever been performed with a Penning trap. Combined with our mass measurements of $^{8,9}$Li we derive a new two-neutron separation energy of 369.15(65) keV: a factor of seven more precise than the best previous value. This new value is a critical ingredient for the determination of the halo charge radius from isotope-shift measurements. We also report results from state-of-the-art atomic-physics calculations using the new mass and extract a new charge radius for $^{11}$Li. This result is a remarkable confluence of nuclear and atomic physics.
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