Deep R-band CCD linear polarimetry collected for fields with lines of sight toward the Lupus I molecular cloud is used to investigate the properties of the magnetic field within this molecular cloud. The observed sample contains about 7000 stars, almost 2000 of them with a polarization signal-to-noise ratio larger than 5. These data cover almost the entire main molecular cloud and also sample two diffuse infrared patches in the neighborhood of Lupus I. The large-scale pattern of the plane-of-sky projection of the magnetic field is perpendicular to the main axis of Lupus I, but parallel to the two diffuse infrared patches. A detailed analysis of our polarization data combined with the Herschel/SPIRE 350{mu}m dust emission map shows that the principal filament of Lupus I is constituted by three main clumps that are acted on by magnetic fields that have different large-scale structural properties. These differences may be the reason for the observed distribution of pre- and protostellar objects along the molecular cloud and the cloud's apparent evolutionary stage. On the other hand, assuming that the magnetic field is composed of large-scale and turbulent components, we find that the latter is rather similar in all three clumps. The estimated plane-of-sky component of the large-scale magnetic field ranges from about 70 to 200{mu}G in these clumps. The intensity increases toward the Galactic plane. The mass-to-magnetic flux ratio is much smaller than unity, implying that Lupus I is magnetically supported on large scales.