The first measurements of double-hadron production in deep-inelastic scattering within the nuclear medium were made with the HERMES spectrometer at DESY HERA using a 27.6 GeV positron beam. By comparing data for deuterium, nitrogen, krypton, and xenon nuclei, the influence of the nuclear medium on the ratio of double-hadron to single-hadron yields was investigated. Nuclear effects on the additional hadron are clearly observed, but with little or no difference among nitrogen, krypton, or xenon, and with smaller magnitude than effects seen on previously measured single-hadron multiplicities. The data are compared with models based on partonic energy loss or prehadronic scattering and with a model based on a purely absorptive treatment of the final-state interactions. Thus, the double-hadron ratio provides an additional tool for studying modifications of hadronization in nuclear matter.