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It is now accepted that long-duration gamma-ray bursts (GRBs) are produced during the collapse of a massive star. The standard 'collapsar' model predicts that a broad-lined and luminous type Ic core-collapse supernova accompanies every long-duration GRB. This association has been confirmed in observations of several nearby GRBs. Here we report that GRB 060505 (ref. 10) and GRB 060614 (ref. 11) were not accompanied by supernova emission down to limits hundreds of times fainter than the archetypal supernova SN 1998bw that accompanied GRB 980425, and fainter than any type Ic supernova ever observed. Multi-band observations of the early afterglows, as well as spectroscopy of the host galaxies, exclude the possibility of significant dust obscuration and show that the bursts originated in actively star-forming regions. The absence of a supernova to such deep limits is qualitatively different from all previous nearby long-duration GRBs and suggests a new phenomenological type of massive stellar death.