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 (SN) accompanies every long-duration GRB. This association has been confirmed in observations of several nearby GRBs. Here we present observations of two nearby long-duration GRBs that challenge this simple view. In the GRBs 060505 and 060614 we demonstrate that no SN emission accompanied these long-duration bursts, down to limits hundreds of times fainter than the archetypal SN 1998bw that accompanied GRB 980425, and fainter than any Type Ic SN 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 star-forming regions. The absence of a SN to such deep limits is qualitatively different from all previous nearby long GRBs and suggests a new phenomenological type of massive stellar death. From the supplementary material: Now we have observed SN-less GRBs in star-forming regions, suggesting that a non-detection of a SN does not preclude a massive progenitor. The position of the GRB, i.e. in a star-forming region or in an older component, may be the only way to discriminate between merging compact objects and massive stars as progenitors. In fact, several host galaxies for short GRBs have been found to be as actively star-forming as some host galaxies of long-duration GRBs. The GRB labels long and short have become synonymous with massive stars and other progenitors. These distinctions may need to be relaxed.