Let q be a prime power with q≡−1 ( mod 4) , and let F be the finite field with q elements and Q the set of nonzero squares in F. Let G=PSL(2,q) be the special linear fractional group on Ω={∞}∪F , the projective line over F, and set V={∞}∪(Q▵(Q+1)▵(Q−1)), V =Ω⧹V , where ▵ denotes the symmetric difference. First, we consider the cardinality of intersections of some translations of Q in F and show [Formula: see text] Next, when 2∉Q, we determine the structure of G _V=G _V , the setwise stabilizer of V or V in G, and show that the design (Ω, V ^G) is a 3-(q+1,(q−3)/2,λ) design, where [Formula: see text] This is a new infinite family of 3-designs.