Clue: Minister losing head meeting American — one on perilous journey (6)
Answer: ICARUS
Explanation: Definition: one on perilous journey
A charade of [v]ICAR ('minister') minus the first letter ('losing head') plus US ('American'). Daedalus and his son Icarus, by the Greek legend, attempted to escape from Crete by flying with wings attached to their arms by wax; Icarus flew too near the sun, the wax melted, and he plummeted into the sea to drown.