Williams, who toured Australia with the Lions in 2001, takes over from 34-year-old Bath hooker Jonathan Humphreys, who has been ruled out because of a shoulder injury.
Humphreys had replaced Swansea forward Colin Charvis as skipper against England last month following Wales' defeat in Italy.
Wales coach Steve Hansen has made two changes from the side beaten 26-9 by England at the Millennium Stadium.
Llanelli fly-half Stephen Jones makes his first start of the championship after recovering from a broken thumb, and is recalled immediately to the side instead of Ceri Sweeney, who does not find a place in the 22.
The other switch is an enforced one in the front row, where Bridgend hooker Gareth Williams makes his full Test debut instead of Humphreys.
Former rugby league star Iestyn Harris keeps his place on the bench, although many will feel that Sweeney is unlucky to drop out following an impressive performance against England when he kicked all of Wales' points.
Hansen has no doubts about Williams' ability to lead Wales against Scotland.
"He is well respected on and off the field, and he leads by example," said Hansen.
"Having a third captain in three games is obviously not an ideal situation, but that is where we are at. Martyn has been a unit leader within the team for quite some time, and now he gets the opportunity to step up.
"There is no doubt that he has been one of our most consistent performers."
Wales team to play Scotland in RBS 6 Nations Championship at Murrayfield on Saturday:
K Morgan (Swansea); R Williams (Cardiff), M Taylor (Swansea), T Shanklin (Saracens), G Thomas (Bridgend); S Jones (Llanelli), G Cooper (Bath); I Thomas (Llanelli), G Williams (Bridgend), B Evans (Swansea), R Sidoli (Pontypridd), S Williams (Northampton), D Jones (Llanelli), M Williams (Cardiff, capt), G Thomas (Bath).
Replacements: M Davies (Pontypridd), G Jenkins (Pontypridd), G Llewellyn (Neath), C Charvis (Swansea), D Peel (Llanelli), I Harris (Cardiff), M Watkins (Llanelli).