Ms Ledsham was last seen at her home in the Dounehill area of Jedburgh in the Scottish Borders at around 11pm on Wednesday, January 20.
The 62-year-old is believed to have left the property in a dark blue Toyota Auris with the registration: SL68 CHV.
On Friday morning, officers confirmed that a body had been discovered near Leaderfoot Viaduct at around 8:20pm the previous evening.
Although a formal identification is yet to take place, police said Ms Ledsham’s family have been informed.
In a statement, Tweed Valley Mountain Rescue said: “Last night the team were called by Police Scotland to assist in the search for a missing person in the Leaderfoot Viaduct area of Melrose.
“14 Team members assisted in the search; in addition, Border Search and Rescue Unit were on standby to continue the search had it been prolonged.
“Sadly, a body was found by search teams on Thursday evening shortly after search activity commenced.
“Our thoughts and condolences are with the family and friends of the missing person,” the statement concluded.
It was the group's seventh callout of the year so far.
Ms Ledsham is around 5’10 tall, with a slim build and straight, collar length red hair.
Appealing for the public’s help to find her on Thursday, Sergeant Phil Winning of Border police, said: "This is completely out of character for Charmain and we are growing increasingly concerned for her safety and wellbeing.”