undertaker

An undertaker is somebody whose job is preparing the dead for burial. Collecting the body, arranging the funeral, carrying the coffin.

The profession now generally prefers funeral director, which covers the wider work of arranging a whole funeral, and undertaker sounds slightly old-fashioned though it is not disrespectful. American English says mortician. The name comes from undertaking the task, in the plain old sense of taking something on. The trade is one of the very few that has never lacked customers.