The Netherlands international is the second man to move to the AmEx this summer, following the acquisition of Adam Lallana from Liverpool on a free transfer.
Brighton boss Graham Potter pointed to the experience Veltman, 28, possesses at the top level as a deciding factor in signing him.
"He has terrific experience from his time with Ajax including the Champions League," he said.
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"He's an excellent defender who can play at right-back and at centre half, but he's also comfortable in possession."
Veltman came through the ranks in the Dutch capital, making over 200 appearances for the club and winning three Eredivisie titles.

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Brighton avoided relegation from the Premier League by seven points and can now look forward to a fourth straight season of top-flight football.
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16/10/2021 AT 16:18