WE PROPOSE THE FIRST PROMPT SEARCHES FOR X-RAY AND UV/OPTICAL COUNTERPARTS TO ICECUBE HIGH-ENERGY (E>~ 30 TEV) LIKELY-COSMIC NEUTRINOS WITH SWIFT. USING NEWLY-INSTALLED HIGH-ENERGY EVENT IDENTIFICATION AND RECONSTRUCTION ALGORITHMS RUNNING AT THE SOUTH POLE WE WILL SELECT UP TO THREE NEUTRINOS WITH HIGH PROBABILITIES OF COSMIC ORIGIN AND GOOD LOCALIZATIONS. EACH TRIGGER WILL BE OBSERVED WITH A MOSAIC OF 19 SWIFT POINTINGS TILING THE NEUTRINO LOCALIZATION TO>50% COMPLETENESS AND 1 KS EXPOSURE WITHIN ROUGHLY 16 HOURS IN SEQUENCE DESIGNED TO RECOVER>50% OF SWIFT GRB X-RAY AFTERGLOWS. DISCOVERY OF EVEN A SINGLE EM COUNTERPART TO A HIGH-ENERGY NEUTRINO WOULD REVOLUTIONIZE OUR UNDERSTANDING OF THE SOURCE POPULATION AND POTENTIALLY REVEAL THE ORIGINS OF THE HIGHEST-ENERGY COSMIC RAYS.