In a state of sleep; also, broadly, resting.
"The baby drifted off to sleep and remained asleep throughout the night while we rested quietly nearby."
In plain English: Asleep means you are sleeping and not awake anymore.
"The baby fell asleep quickly after drinking his milk."
Usage: Use asleep only as an adjective placed after the noun it modifies or before another verb form, such as in "the baby is fast asleep." Do not use it predicatively with linking verbs like be unless followed by an adverbial phrase indicating location or manner.
in the sleep of death
"The ancient king lay asleep in his stone sarcophagus, never to wake again."
The word asleep comes from Middle English aslepe, which combined the prefix meaning "in or on" with the root for sleeping. This formation directly described the state of being in a deep slumber.