We report measurements of record-setting intensities of cosmic-ray nuclei from C to Fe, made with the Cosmic Ray Isotope Spectrometer carried on the Advanced Composition Explorer in orbit about the inner Sun-Earth Lagrangian point. In the energy interval from ∼70 to ∼450 MeV nucleon −1 , near the peak in the near-Earth cosmic-ray spectrum, the measured intensities of major species from C to Fe were each 20%-26% greater in late 2009 than in the 1997-1998 minimum and previous solar minima of the space age (1957-1997). The elevated intensities reported here and also at neutron monitor energies were undoubtedly due to several unusual aspects of the solar cycle 23/24 minimum, including record-low interplanetary magnetic field (IMF) intensities, an extended period of reduced IMF turbulence, reduced solar-wind dynamic pressure, and extremely low solar activity during an extended solar minimum. The estimated parallel diffusion coefficient for cosmic-ray transport based on measured solar-wind properties was 44% greater in 2009 than in the 1997-1998 solar-minimum period. In addition, the weaker IMF should result in higher cosmic-ray drift velocities. Cosmic-ray intensity variations at 1 AU are found to lag IMF variations by 2-3 solar rotations, indicating that significant solar modulation occurs inside ∼20 AU, consistent with earlier galactic cosmic-ray radial-gradient measurements. In 2010, the intensities suddenly decreased to 1997 levels following increases in solar activity and in the inclination of the heliospheric current sheet. We describe the conditions that gave cosmic rays greater access to the inner solar system and discuss some of their implications.