  We apply strong magnetic fields of H=28.5 \sim 43 T to suppress
superconductivity (SC) in the cuprates Bi_{2}Sr_{2-x}La_xCuO_{6+\delta}
(x=0.65, 0.40, 0.25, 0.15 and 0), and investigate the low temperature (T)
normal state by ^{63}Cu nuclear spin-lattice relaxation rate (1/T_1)
measurements. We find that the pseudogap (PG) phase persists deep inside the
overdoped region but terminates at x \sim 0.05 that corresponds to the hole
doping concentration of approximately 0.21. Beyond this critical point, the
normal state is a Fermi liquid characterized by the T_1T=const relation. A
comparison of the superconducting state with the H-induced normal state in the
x=0.40 (T_c = 32 K) sample indicates that there remains substantial part of the
Fermi surface even in the fully-developed PG state, which suggests that the PG
and SC are coexisting matters.
