Conversion from neutron stars to strange stars as a possible mechanism of c
osmological gamma-ray bursts (GRB s) has been discussed in previous works,
although the existence of strange stars is still an open question. On the b
asis of this mechanism, we here outline an explanation of the connection be
tween supernovae (SNe) and GRBs, which has got increasing evidence recently
. An asymmetric but normal SN explosion leaves a massive (greater than or e
qual to 1.8M(circle dot)) and rapidly rotating neutron star, which then con
verts to a strange star few days later, due to its rapid spin down. The acc
ompanied fireball, which can be accelerated to ultra-relativistic velocity
(To similar to 100) due to the very low baryon contamination of the strange
star, flows out along the direction of the high-velocity SN jet and subseq
uently produces a GRB and the following low energy afterglows by interactin
g with the surrounding stellar wind. We will also expect a very luminous su
pernova like SN1998bw, if a large fraction of the conversion energy finally
turns into the kinetic energy of the supernova ejecta.