Green’s Dictionary of Slang

(drugs) the Federal drug habilitation hospital at Lexington, Kentucky.

1953		W. Brown Monkey On My Back (1954) 159: Would he be willing to take the cure again? No. He’d kill himself before he went back to Lex.
1960		C. Cooper Jr Scene (1996) 111: I am not a junkie. Lex is for junkies. I ain’t hooked.
1971		E.E. Landy Underground Dict. (1972).