The 24-year-old has signed a four-year contract and joins the Spanish fee for a fee believed to be around €7million.
“I wanted to sign for one of the best club’s in Europe,” he told their official website.
“I’m happy to be a part of this club.”
Alderweireld has also been linked with the likes of Bayer Leverkusen and Liverpool this summer.
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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.