The rise of pan-resistant bacteria is creating an urgent need for structurally novel antibiotics. AI methods can discover new antibiotics, but existing methods have significant limitations. Property prediction models, which evaluate molecules one-by-one for a given property, scale poorly to large chemical spaces. Generative models, which directly design molecules, rapidly explore vast chemical spaces but generate molecules that are challenging to synthesize. Here, we introduce SyntheMol, a generative model that designs easily synthesizable compounds from a chemical space of 30 billion molecules. We apply SyntheMol to design molecules that inhibit the growth of Acinetobacter baumannii, a burdensome bacterial pathogen. We synthesize 58 generated molecules and experimentally validate them, with six structurally novel molecules demonstrating potent activity against A. baumannii and several other phylogenetically diverse bacterial pathogens.