The IceCube Neutrino Observatory is a cubic kilometer neutrino telescope located at the geographic South Pole which can observe neutrinos at energies from GeV to above a PeV. Now that IceCube has realized its primary goal, the detection of high-energy astrophysical neutrinos, the task remains to identify the sources of these neutrinos. Observing neutrinos in coincidence with transient astrophysical events would be a smoking gun for hadronic acceleration and provide other valuable insights into the nature of the sources. This contribution discusses a new approach to investigate transient phenomena by performing a rapid follow-up analysis. This is done by searching for neutrinos in coincidence with detections by other observatories so that the results can be distributed rapidly to the rest of the astronomical community. In order to accomplish this with minimum delay, the data must be processed immediately at the South Pole and be transmitted to the Northern Hemisphere via the Iridium satellite network. The data processing pipelines, maximum likelihood analysis, and sensitivity of this analysis will be discussed.