An apparatus is used to support and move vertically an erected form assembly during the construction of a building with a plurality of stories. The stories include a lower story and an upper story. Each of the lower and upper stories is formed previously with a concrete wall structure that has surrounding walls formed integrally with reinforced horizontal beams which confine an interior space. The apparatus includes a form support body that can extend across the interior space of the upper story and that has a top side on which the erected form assembly can rest. A plurality of legs extend downwardly from the form support body. Each leg has a lower portion associated with a clamping unit in order to fix releasably the leg to one of the reinforced horizontal beams of the lower story. A horizontal support bar is located below and is spaced from the form support body. The horizontal support bar can extend across the interior space of the lower story and has two ends to be mounted releasably on the reinforced horizontal beams of the lower story. A moving unit, which is connected to the form support body and the horizontal support bar, is used for moving vertically the horizontal support bar with respect to the form support body.