X: I can not give you the accurate date of my arrival yet.
O: I can not give you the exact date of my arrival yet.
Accurate is mainly used (1) to describe something said or written that contains no errors: "That movie is historically accurate." (2) to describe something that produces no errors. "I'm not sure my bathroom scale is accurate."
When you mean "correct and as detailed as possible" use exact. "The exact time is 7:54."
