The first appearance of Batman as a character was in Detective Comics #27, though he was called "The Bat-Man" at the time, in a story written by Bill Finger and drawn by Bob Kane. Note that as early as the third Batman story, someone else was doing the writing, though Finger would continue to write Batman for Detective Comics on and off for a while. Some or all of the artistry was done by Bob Kane for most of this period (though again, duties were shared with other artists).
By the early 1940s (1942-1943) Kane and especially Finger did fewer and fewer Detective Comics Batman stories. Their last Detective Comics issue together was Detective Comics #190 (December, 1952), but by then they were collaborating once every couple of years at best. The two would continue to contribute for a while, and Finger in particular came back in the early 50s to become a fairly regular writer (well after most people would claim that his "original" Batman was no longer being featured).
Around the time their work on Detective Comics began dropping off, the solo Batman series (June 1940) was starting up, with Kane and Finger also doing the early issues. They would continue to write/draw off and on for this series well into the 1950's.
As best as I can tell, the absolute last time Bill Finger wrote a Batman story was Detective Comics #328 (June, 1964), while Bob Kane's last issue was Detective Comics #228 (February, 1956).
Of course, the character itself, the "Golden Age" Batman, would live on past its co-creators. This version of the character "died" in Adventure Comics #462, in April of 1979. By that time, he had been established (in the Justice League of America series) as being the Earth-Two Batman, and would reappear on very rare occasions as Earth-Two's Bruce Wayne in the crossover events.
There is no clear point in either Detective Comics nor the solo Batman series when Earth-Two Batman obviously becomes the later Earth-One Batman. Most people would point to Superman #76, in June of 1952, as the first appearance that is definitively Earth-One's Batman, but the other series likely continued to include the Golden Age Batman for many issues afterward. The two most popular theories for the "last appearances" of Earth-Two Batman in his solo series are in Batman #80 (December, 1953) or Batman #95 (October, 1955). For Detective Comics, it is usually placed in either Detective Comics #202 (December, 1953) or Detective Comics #224 (October, 1955)