The bisection method was applied to localize a solution of Kepler's equation in an extended form where the Newton-Raphson and some other methods are not always stable. This localizer was combined with a variation of the Newton-Raphson method where trigonometric functions are evaluated by Taylor series expansions. As a result, we developed a procedure solving the extended Kepler's equation. It is roughly twice as fast as Halley' s method and more for other existing schemes. This significantly speeds up Encke' s method applied to orbital elements (Fuknshima 1996 a). Also, even in solving the original Kepler' s equation, it is 20% faster than the author's former method (Fukushima l996b).