When was the first official game arranged between two baseball clubs?

The first official game arranged between two baseball clubs was in 1846.

At Elysian Fields in Hoboken, New Jersey, the first official game prearranged between two separate clubs (the Knickerbockers and the New York Nine) was played using the rules developed by Cartwright, Adams, and the Knickerbockers. The Knickerbockers lost the four-inning game 23-1.