  Long-duration gamma-ray bursts (GRBs) are thought to be connected to luminous
and energetic supernovae (SNe), called hypernovae (HNe), resulting from the
black-hole (BH) forming collapse of massive stars. For recent nearby
GRBs~060505 and 060614, however, the expected SNe have not been detected. The
upper limits to the SN brightness are about 100 times fainter than
GRB-associated HNe (GRB-HNe), corresponding to the upper limits to the ejected
$^{56}$Ni masses of $M({\rm ^{56}Ni})\sim 10^{-3}M_\odot$. SNe with a small
amount of $^{56}$Ni ejection are observed as faint Type II SNe. HNe and faint
SNe are thought to be responsible for the formation of extremely metal-poor
(EMP) stars. In this Letter, a relativistic jet-induced BH forming explosion of
the 40 $M_\odot$ star is investigated and hydrodynamic and nucleosynthetic
models are presented. These models can explain both GRB-HNe and GRBs without
bright SNe in a unified manner. Their connection to EMP stars is also
discussed. We suggest that GRBs without bright SNe are likely to synthesize
$\Mni\sim 10^{-4}$ to $10^{-3}M_\odot$ or $\sim 10^{-6}M_\odot$.
