Solids with strong electron correlations generally develop exotic phases of electron matter at low temperatures(1-3). Among such systems, the heavy-fermion semimetal URu(2)Si(2) exhibits an enigmatic transition at T(0) = 17.5 K to a 'hidden-order' state for which the order parameter remains unknown after ;23 years of intense research(4,5). Various experiments point to the reconstruction and partial gapping of the Fermi surface when the hidden order establishes(6-14). However, up to now, the question of how this transition affects the electronic states at the Fermi surface has not been directly addressed by a spectroscopic probe. Here we show, using angle-resolved photoemission spectroscopy, that a band of heavy quasiparticles drops below the Fermi level on the transition to the hidden-order state. Our data provide the first direct evidence of a large reorganization of the electronic structure across the Fermi surface of URu(2)Si(2) occurring during this transition, and unveil a new kind of Fermi-surface instability in correlated electron systems.