Schick joins the Bundesliga club until 30 June 2020, for a fixed fee of €3.5 million.
A further €500,000 in performance-related payments could also be made, depending on certain targets being met by both the player and the German club.
As part of the overall agreement, should Leipzig qualify for the 2020-2021 UEFA Champions League they will have the option to make the transfer permanent - for a set fee of €29m.
Otherwise the club have an option to make the deal permanent for €28m at the end of the season.
The Czech Republic international, 23, joined Roma in the summer of 2017.
He went on to make 57 appearances for the club in all competitions, scoring eight goals.
The club would like to wish Patrik the best of luck for his time in Germany.
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