Reduced pipe dreams are combinatorial objects that encode some of the algebraic, enumerative, geometric, and probabilistic properties of Schubert and Grothendieck polynomials. They were introduced in 1993 by Bergeron and Billey, who showed that the set of all reduced pipe dreams for a fixed permutation w has a natural poset structure, with covering relations given by simple local operations called chute and ladder moves. In 2011, Rubey generalized chute and ladder moves on the set of reduced pipe dreams for a permutation w, and he conjectured that the induced poset on reduced pipe dreams is a lattice. We prove this conjecture and give simple recursive formulas for joins and meets in Rubey's lattice. This talk is based on joint work with Sara Billey and Clare Minnerath.