It was savaged in the field to the rear of Marmion Road between 10am and 2pm on Sunday, November 12.
Inspector Carol Wood, of Hawick police station, said: “This is the third reported incident of sheep-worrying in the Hawick area in the past two months and the second for this farmer.
“The sheep was found with such serious injuries it had to be put down, and the animal that attacked it would likely have blood on its face.
“I would remind dog owners that it is your responsibility to ensure that your dog is kept under suitable control at all times and would ask anyone with information about this incident to contact police as soon as possible.”
Potential witnesses are asked to call Hawick police station via 101, quoting incident number 2,183 of November 12.
Alternatively, information can be reported anonymously to the charity Crimestoppers on 0800 555111.