The Canterbury-Bankstown Bulldogs and Des Hasler part ways, effectively immediately.
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Kaufusi has signed for next season with the option of an additional year while Welch has put pen to paper on a new two-year deal to remain in Melbourne.
The re-signing of the young props is yet another significant boost for Storm's forward pack.
Welch, 22, has played 33 games in the purple jersey including last year's NRL Grand Final.
Over the last two years his aggressive style of play has proved a real asset for Melbourne.
This season has seen Kaufusi break into Craig Bellamy's starting side.
The 24-year-old Tongan international has played every minute, of every game this season, taking his opportunity in the second row with both hands.
Both players have been recognised at State of Origin level, with Kaufusi and Welch earning selection in Queensland's Emerging Origin squad.
Kaufusi was also named as 18th man for the Maroons in Game III of the 2016 State of Origin Series.
Storm Football Director Frank Ponissi said the Club was delighted to secure the exciting pair: "Felise and Christian are a big part of what we are building at this Club for the future.
"Both players are growing into their own at NRL level and we are looking forward to seeing them take that next, significant step in their development.
"To have Felise and Christian re-signed, along with Jesse, Kenny, Joe and Dale, we are confident our forward pack has a strong foundation for years to come."