keep_one's_mouth_shut
1. [ verb ] refrain from divulging sensitive information; keep quiet about confidential information
Synonyms: shut_one's_mouth keep_quiet
Examples:

"Don't tell him any secrets--he cannot keep his mouth shut!"

Related terms: tattle
Similar spelling:   keep_one's_distance
  keep_one's_eyes_skinned
  keep_one's_eyes_off