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Despite the scoreline it was the Borders who had bossed the second period with Ross Ford outstanding for the visitors.
Andy Newman, James Bater and Richard Pugh crossed for first half tries for the Ospreys who were short several test players, and by half time they had built a 27-3 lead.
However the Borders produced a much better performance in the second half keeping the league leaders on the back foot for long periods with Charlie Hore adding a converted try to his first half penalty.
Shaun Connor converted Millward’s late try, to add to the three first half conversions and two penalties.