We report the first observations of beam losses due to bound-free pair production at the interaction point of a heavy-ion collider. This process is expected to be a major luminosity limit for the Large Hadron Collider (LHC) when it operates with 208Pb82+ ions because the localized energy deposition by the lost ions may quench superconducting magnet coils. Measurements were performed at the Relativistic Heavy Ion Collider (RHIC) during operation with 100 GeV/nucleon 63Cu29+ ions. At RHIC, the rate, energy and magnetic field are low enough so that magnet quenching is not an issue. The hadronic showers produced when the single-electron ions struck the RHIC beampipe were observed using an array of photodiodes. The measurement confirms the order of magnitude of the theoretical cross section previously calculated by others.