Weir, whose three previous caps have been won from the bench, replaces his club-mate Ruaridh Jackson in the pivotal position, while Euan Murray returns to the side after missing last month's 12-8 win over Ireland on religious grounds.
Weir said: "It's a moment I'll cherish and hopefully I can do the coaches justice for giving me the start.
"It will be a hugely proud moment, not only for me but for my family and friends. I'll get to share a great achievement with them and the Scottish nation as well.
"When I came on against Ireland I just wanted to bring a real energy, put a wee bit of pressure on Ireland and get a bit of territory. Our defence for the majority of the game had been outstanding, so I knew our defence could bring us points.
"I also wanted to try and get the guys going in attack, so I felt I did all right there."
Former British and Irish Lion and devout Christian Murray sat out the triumph over Declan Kidney's men for religious reasons.
However, Scotland interim head coach Scott Johnson has restored him to his starting line-up at the tighthead berth with Geoff Cross moving to the bench.
Meanwhile, Ryan Wilson, the 23-year-old Glasgow Warriors number eight who was man of the match in Scotland A's victory over England Saxons last month, is also introduced to a match-day 23 for the first time.
Wilson is named instead of David Denton who sustained a knee injury when playing for Edinburgh against Scarlets at the weekend.
Johnson said: "Selection is tough just now but that's a good thing as we increase competition for places.
"Duncan did well when he came on against Ireland and he got some decent game time for Glasgow at the weekend. He's a talent with a really good work ethic and a thirst to improve, which I like a lot.
"Ruaridh remains a key part of our squad but he knows with Tom Heathcote proving a match-winner for Bath and kicking goals from here, there and everywhere, that he needs to up his game too.
"Geoff Cross was a standout against Ireland and it's up to Euan, who fronted up big time against Italy, to raise the bar again.
"Wales will be going for their fifth successive away victory in the championship and that would be a record for them but our focus is on ensuring we win the contact zone, continue to improve and make it three wins on the bounce."
Johnson has made an immediate impact since taking over from Andy Robinson.
The squad that trudged off with the wooden spoon last year has now picked up two wins from three and are genuine Six Nations title contenders.
Weir believes it is Johnson's attention to detail, especially with the half-backs, which has played a large part in that fact.
He said: "Johnno drills the stand-offs pretty hard. He wants us to know our goals, our movement onto the ball and our leadership of the team. He also wants us to know the correct calls and also to listen for our outside players and what they can see before relaying that back to everyone else and making sure we exploit the chances we get.
"We were disappointing against England but we played some great rugby against Italy. We didn't have much good ball to attack against Ireland but our defence was superb and that was what won us the game.
"We can still iron out the creases and hopefully against Wales people will see a rounded-off performance."
Meanwhile, Gethin Jenkins will miss Wales' RBS 6 Nations clash against Scotland at Murrayfield on Saturday.
The Toulon prop, who has won 97 caps, suffered a calf muscle injury during the 26-9 victory over Italy in Rome 10 days ago and is replaced by Bath forward Paul James.
Two other changes from that game see flanker Sam Warburton - last season's Grand Slam captain - return to the starting line-up instead of Justin Tipuric, while lock Alun-Wyn Jones replaces Andrew Coombs.
But even though Warburton has been recalled - he suffered a shoulder injury earlier in the championship - his back-row colleague Ryan Jones remains as skipper.
Jones, who has captained Wales a record 31 times, took over from Warburton when Wales beat France in Paris, and he unsurprisingly keeps the job this weekend.
Wales' interim head coach Rob Howley retains the same back division for a fourth successive game, but three changes on the bench from Rome see Cardiff Blues prop Scott Andrews come in for James, with Coombs and Tipuric swapping places with Jones and Warburton respectively.
"Ryan has led the team to two wins and deserves to captain the side against Scotland," Howley said.
"We have spoken a lot about momentum in this tournament, and we go into the game with two wins under our belt, but Scotland are in the same position and it's going to be a good challenge."
It will be Alun-Wyn Jones' first Test start since he suffered a shoulder injury against Argentina last November and subsequently underwent surgery.
It is a tough call on Newport Gwent Dragons forward Coombs, whose performances in the Six Nations so far have impressed, but Jones offers considerable experience, having won almost 70 caps.
Wales remain in contention to successfully defend the Six Nations title, although unbeaten England remain tournament favourites.
Howley added: "We have had two good, hard-fought away victories, but we know we will have to be at our best again to come away from Murrayfield with the result.
"It's disappointing for Gethin to miss out, but Paul has impressed and made an impact in each game he has played.
"We are bringing in a wealth of experience with Alun-Wyn and Sam, and they both performed well off the bench in Rome."
Scotland team to play Wales in the RBS 6 Nations Championship on Saturday, March 9, at Murrayfield (kick-off 2.30pm):
S Hogg (Glasgow); S Maitland (Glasgow), S Lamont (Glasgow), M Scott (Edinburgh), T Visser (Edinburgh); D Weir (Glasgow), G Laidlaw (Edinburgh); R Grant (Glasgow), R Ford (Edinburgh), E Murray (Worcester), R Gray (Sale), J Hamilton (Gloucester), R Harley (Glasgow), K Brown (Saracens), J Beattie (Montpellier).
Replacements: D Hall (Glasgow), M Low (Glasgow), G Cross (Edinburgh), A Kellock (Glasgow), R Wilson (Glasgow), H Pyrgos (Glasgow), R Jackson (Glasgow), M Evans (Castres).
Wales team to play Scotland in the RBS 6 Nations Championship on Saturday, March 9 at Murrayfield (kick-off 2.30pm):
L Halfpenny (Cardiff Blues); A Cuthbert (Cardiff Blues), J Davies (Scarlets), J Roberts (Cardiff Blues); G North (Scarlets); D Biggar (Ospreys), M Phillips (Bayonne); P James (Bath), R Hibbard (Ospreys), A Jones (Ospreys), A-W Jones (Ospreys), I Evans (Ospreys), R Jones (Ospreys, capt), S Warburton (Cardiff Blues), T Faletau (Newport Gwent Dragons).
Replacements: K Owens (Scarlets), S Andrews (Cardiff Blues), C Mitchell (Exeter), A Coombs (Newport Gwent Dragons), J Tipuric (Ospreys), L Williams (Cardiff Blues), J Hook (Perpignan), S Williams (Scarlets).