A classical branching theorem of Weyl describes how an irreducible representation of compact U(n+1)U(n+1) decomposes when restricted to U(n)U(n). The local Gan-Gross-Prasad conjecture provides a conjectural extension to the setting of representations of noncompact unitary groups lying in a generic L-packet. We prove this conjecture. Previously Beuzart-Plessis proved the ''multiplicity one in a Vogan packet'' part of the conjecture for tempered L-packets using the local trace formula approach initiated by Waldspurger. Our proof uses theta lifts instead, and is independent of the trace formula argument.