  Electrical control of magnetism is a long-standing goal in physics and
technology, recently developed electrolyte gating techniques providing a
promising route to realization. Validating a recent theoretical prediction,
here we demonstrate large enhancement of electrostatic modulation of
ferromagnetic order in ion-gel-gated ultrathin
La$_{0.5}$Sr$_{0.5}$CoO$_{3-\delta}$ by thickness-tuning to the brink of a
magnetic percolation transition. Application of only 3-4 V then drives a
transition from a short-range-ordered insulator to a robust long-range
ferromagnetic metal, realizing giant electrostatic Curie temperature modulation
over a 150 K window. In operando polarized neutron reflectometry confirms
gate-controlled ferromagnetism, also demonstrating unusually deep penetration
of induced magnetization, in further agreement with theory.
