The A-B-A block copolymers were prepared by the ring opening polymerization of L-lactide by the polypropylene glycol (PPG; Mw/Mn=1.1, Mn=4000) with $Me_3Al-H_2O$ as the catalyst. When the feed ratios of PPG were over 5 and 10 wt% relative to L-lactide, the polymerization of L-lactide took place from the PPG hydroxyl terminals to give the desired A-B-A block copolymers in high yields. The resulting molecular weights were in good agreement with the values estimated from the monomer conversions and the feed ratios of PPG. At the lower feed ratios of PPG, the poly(L-lactic acid) (PLLA) homopolymer was formed along with the block copolymers. The block copolymers were melt-spun by the conventional method and the fibers obtained were extended by drawing at $60^{circ}C$. At the same draw ratio, the modulus of the fibers was decreased with increasing PPG content in the block copolymers. The fibers of the PLLA-PPG-PLLA block copolymers with different contents of PPG were subjected to degradation in vitro. Upon immersion in a phosphate buffer solution (pH=7.4), the fibers showed a time-dependent decrease in tensile strength with accompanying surface erosion. The degradation rate of this fiber was much higher than that of PLLA fiber. It was therefore suggested that the block copolymers comprising PLLA and PPG, have high potential as the biodegradable suture with improved flexibility and biodegradability.