Dense, well-ordered material states of proteins form the amyloid fibres that are indicative of neurodegenerative disease in the brain. In addition to forming amyloid fibres, some of these proteins comprise condensates that are initially liquid-like but evolve to a glassy, solid-like phase. Fibre growth requires a large supply of monomers, and so it is often speculated that fibres emerge from within a dense condensate as it ages and its microscopic dynamics slow into a glassy state. Here we show that, in the model system Fused in Sarcoma (FUS), fibres grow overwhelmingly in the dilute phase surrounding the condensates; concurrently, the condensates evolve to a glassy arrested solid. The resulting protein fibres and glassy condensates are both distinct solid-like phases that coexist but do not directly interconvert. These findings suggest that there may be two competitive ageing pathways in FUS condensation that are linked through phase separation kinetics.