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The natural conductivity of as-grown ZnO is $n$-type. It has been challenging to produce stable $p$-type material, which has delayed possible technological applications. In this work, the conductivity transformation from $n$- to $p$-type ZnO films deposited by sputtering was followed as a function of copper and tellurium concentrations and of the substrate temperature during growth (room temperature, $150{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\circ}}\mathrm{C}, 250{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\circ}}\mathrm{C}$, and $350{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\circ}}\mathrm{C}$). The nominal codopant concentrations ranged from 1 to 12 at %. For the minimum concentration, compensation effects yielded highly resistive ZnO. Nonetheless, by tailoring the concentration of Cu and Te, it was possible to vary the resistivity $({10}^{3}--{10}^{\ensuremath{-}2}\phantom{\rule{0.16em}{0ex}}\mathrm{\ensuremath{\Omega}}\phantom{\rule{0.16em}{0ex}}\mathrm{cm})$, mobility $(\ensuremath{\sim}{10}^{\ensuremath{-}2}--{10}^{\ensuremath{\circ}}\phantom{\rule{0.16em}{0ex}}\mathrm{c}{\mathrm{m}}^{2}/\mathrm{V}\phantom{\rule{0.16em}{0ex}}\mathrm{s})$, and free-hole density $({10}^{15}--{10}^{20}\phantom{\rule{0.16em}{0ex}}\mathrm{c}{\mathrm{m}}^{\ensuremath{-}3})$ of $p$-type ZnO grown at $250{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{\circ}}\mathrm{C}$. Besides modifying the electrical properties, codoping changed the host band structure significantly, producing a band-gap renormalization from 3.2 eV (UV) to 1.8 eV (red). This control over the band gap is advantageous for applications where controllable photon absorption or emission are sought. Experimentally, films were stable for a period of at least six months. Band-gap engineered $p$-type ZnO:(Cu+Te) films open the possibility for the fabrication of all-ZnO optoelectronic devices such as homojunction solar cells and/or light-emitting diodes.