It is now accepted that long-duration γ-ray bursts (GRBs) are produced during the collapse of a massive star$^1,2$. The standard `collapsar’ model$^3$ predicts that a broad-lined and luminous type Ic core-collapse supernova accompanies every long-duration GRB$^4$. This association has been confirmed in observations of several nearby GRBs$^5-9$. Here we report that GRB060505 (ref. 10) and GRB060614 (ref. 11) were not accompanied by supernova emission down to limits hundreds of times fainter than the archetypal supernova SN1998bw that accompanied GRB980425, and fainter than any type Ic supernova ever observed$^12$. 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.