Emergency services were called to a property on McLagan Drive following reports of an incident at around 7.20am on Thursday.
A 35-year-old woman was pronounced dead at the scene. She has since been named as Stacey Hogg from the area.
A 39-year-old man has been arrested and charged in connection with her death. He is due to appear at Selkirk Sheriff Court (today) Friday, June 26.