Ian Carruthers, 31, was reported missing from the town of Annan in Dumfries and Galloway, on January 17.
Police Scotland confirmed that a body discovered on Friday morning on the Solway Firth shoreline at Portling, near Dalbeattie, was that of Mr Carruthers.
The force said that a 25-year-old woman and a 26-year-old man have been arrested and are expected to appear at Dumfries Sheriff Court today.
The woman, from Annan, will appear in relation to the wasting of police time in connection with the ongoing inquiry into the circumstances surrounding the disappearance of Mr Carruthers.
The man, from Longtown in Cumbria, will appear in relation to drug supply offences.