In vitro studies of pure tubulin have suggested that tubulin heterodimers in cells assemble into B-lattice microtubules, where the 8-nm dimers in adjacent protofilaments are staggered by 0.9 nm. This arrangement requires the tube to close by forming a seam with an A-lattice, in which the protofilaments are staggered by 4.9 nm. Here we show that Mal3, an EB1 family tip-tracking protein, drives tubulin to assemble in vitro into exclusively 13-protofilament microtubules with a high proportion of A-lattice protofilament contacts. We present a three-dimensional cryo-EM reconstruction of a purely A-lattice microtubule decorated with Mal3, in which Mal3 occupies the groove between protofilaments and associates closely with one tubulin monomer. We propose that Mal3 promotes assembly by binding to freshly formed tubulin polymer and particularly favors any with A-lattice arrangement. These results reopen the question of microtubule structure in cells.