Abstract: The present invention provides an outer wall construction apparatus for a building, that is, an outer wall construction apparatus for constructing outer wall of a building having slab between floors or wall surface using stone or metal plate structures including first bracket fixed through anchor bolts inserted into the side of the slab or the wall surface along the side of the slab or the wall surface; fixation beam fixed to the first bracket, substantially vertically projected outward from the side of the slab or the wall surface; horizontal beam fixed to the fixing beam to be horizontally arranged; vertical beam fixed to outward side of the horizontal beam to be vertically arranged, having a hollow along the longitudinal direction thereof and an opening part in the shape of slit along the longitudinal direction of the outer surface thereof; second bracket comprising fixation end fixed tightly to the vertical beam along the outer surface thereof through first bolt of which the head has short side narrower t