In a cleaving apparatus (1) for a glass film, an initial crack (10), which is formed at a leading end portion of a preset cleaving line (7) of a glass film (G), is propagated along the preset cleaving line (7) by a thermal stress generated in the glass film (G) through localized heating performed along the preset cleaving line (7) and cooling of a heated region resulting from the localized heating. At this time, a resin sheet (R) having higher flexibility than the glass film (G) is arranged in a cleaving region, and the resin sheet (R) is floated by blowing a gas on a lower surface of the resin sheet (R) by a floating unit (3). Then, a preset cleaving portion of the glass film (G) including the preset cleaving line (7) is lifted and supported while being covered with the floated resin sheet (R) from below, and in this state, the glass film (G) is cleaved.