Motivated by the duplication-correcting problem for data storage in live DNA, we study the construction of constant-weight codes in $\ell_1$-metric. By using packings and group divisible designs in combinatorial design theory, we give constructions of optimal codes over non-negative integers and optimal ternary codes with $\ell_1$-weight $w\leq 4$ for all possible distances. In general, we derive the size of the largest ternary code with constant weight $w$ and distance $2w-2$ for sufficiently large length $n$ satisfying $n\equiv 1,w,-w+2,-2w+3\pmod{w(w-1)}$.