We present the first simultaneous infrared (IR) and visible spectropolarimetric observations of a solar internetwork region. The Fe I lines at 6301.5, 6302.5, 15648, and 15652 Å were observed, with a lag of only 1 minute, using highly sensitive spectropolarimeters operated in two different telescopes (Vacuum Tower Telescope and THEMIS at the Observatorio del Teide). Some 30% of the observed region shows IR and visible Stokes V signals above noise. These polarization signals indicate the presence of kilogauss (kG) magnetic field strengths (traced by the visible lines) coexisting with sub-kG fields (traced by the infrared lines). In addition, one-quarter of the pixels with signal have visible and IR Stokes V profiles with opposite polarity. We estimate the probability density function of finding each longitudinal magnetic field strength in the region. It has a tail of kG field strengths that accounts for most of the (unsigned) magnetic flux of the region.