Previously, people could get married at 16 or 17 as long as they had parental consent.
The new law also applies to cultural or religious marriages that are not registered with the local council.
Conservative MP Pauline Latham, who introduced the bill to Parliament, said it would "transform the life chances of many girls".
Under the new rules, children will not face penalties but adults who facilitate their marriage could face up to seven years in jail and a fine.
This would include adults who took children abroad to carry out the marriage.