A deal was agreed earlier in February to take the 26-year-old playmaker to Stamford Bridge for £37 million, subject to smoothing out personal terms.
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"I am delighted and proud to have signed for such a huge club as Chelsea," Ziyech told the club's website.
"I am looking forward to next season and hope we can achieve great things together."
Lampard 'delighted' to bring Ziyech to Chelsea
Ziyech played a starring role as Ajax reached the Champions League semi-finals in 2019.
But the Eredivisie leaders have struggled in Europe this season after losing Frenkie De Jong and Matthijs de Ligt to Barcelona and Juventus respectively, with reports now circulating that Ziyech's move could spark an exodus of Ajax's top talent.
Chelsea director Marina Granovskaia added: "We are delighted Hakim will be joining us in the summer, having been a key target of the club’s in this last window. He has consistently been one of the most dangerous attacking players in Europe in recent years, which we saw first-hand in our two games against Ajax in the Champions League.
"We wish Hakim and Ajax all the best for the rest of the season and look forward to welcoming him to Stamford Bridge in the summer."
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