We are interested in knowing what type of manifolds are obtained by doing Dehn surgery on closed pure 3-braids in S3. In particular, we want to determine when we get S3 by surgery on such a link. We consider links which are small closed pure 3-braids; these are the closure of 3-braids of the form (σ12e1)(σ22f1)(σ2σ1σ2)2e, where σ1, σ2 are the generators of the 3-braid group and e1, f1, e are integers. We study Dehn surgeries on these links, and determine exactly which ones admit an integral surgery producing the 3-sphere. This is equivalent to determining the surgeries of some type on a certain six component link L that produce S3. The link L is strongly invertible and its exterior double branch covers a certain configuration of arcs and spheres, which we call the hexatangle. Our problem is equivalent to determine which fillings of the spheres by integral tangles produce the trivial knot, which is what we explicitly solve. This hexatangle is a generalization of the pentangle, which is studied in [C.McA. Gordon, J. Luecke, Non-integral toroidal Dehn surgeries, Comm. Anal. Geom. 12 (2004) 417–485].