The left-arm bowler is available for all formats this summer, subject to his international commitments with Scotland.
The 22-year-old has played more than 50 times for Scotland and has taken more than 50 wickets since his debut in 2015.
Watt has taken 29 T20 International wickets and 22 wickets from 18 One-Day Internationals, including 3-55 in Scotland’s historic six-run win over England in June last year.
His performances for Scotland saw Lancashire acquire his services and he featured in four Vitality Blast games for the Red Rose County.
Watt will be available for Derbyshire’s season opener against Durham, starting on April 5.
Derbyshire head of cricket Dave Houghton told the club’s website: “It’s great to have Mark here at Derbyshire. He’s a clever bowler who bowls tightly and has excellent white ball skills.
“He’s still young but has got plenty of international experience already and I’m pleased to get another talented cricketer on board and continue our relationship with Cricket Scotland.”
Watt added: “I’m really looking forward to getting the season started and hopefully I can take my opportunities and do well for the club.
“There are experienced coaches and a good squad here, so I’ll be trying to learn and improve my game in every aspect.”