The energy spectra of cosmic-ray antiprotons and protons near solar minimum were precisely measured with BESS-Polar II (Balloon-borne Experiment with a Superconducting Spectrometer) during a long-duration flight over Antarctica in December 2007 and January 2008. The upper-TOF (UTOF) and lower-TOF (LTOF) scintillator hodoscopes measure the charge and velocity of incident particles. A thin plastic scintillator middle-TOF (MTOF) hodoscope is installed on the lower surface of the magnet bore to measure low-energy particles that cannot reach the LTOF. The MTOF further lowers the threshold energy to about 120 MeV for antiproton or proton measurements. We report absolute spectra of the cosmic-ray antiproton in the range 0.12 to 0.4 GeV and the antiproton/proton ratio calculated with UTOF-MTOF trigger events. These new results are independent of the UTOF-LTOF triggered antiproton spectrum published in 2012 and proton
spectrum published in 2016.