The weakly reinforced random walk (WRRW) on the one-dimensional integer lattice Z starts from the origin of the lattice and at each step it jumps to a neighbouring site, the probability of jumping along a bond being proportional to w{number of previous jumps along that lattice bond}, where w: N\toR_+, with w(n)\sim n^\alpha for large n, and \alpha\in(0,1) is a fixed parameter. We prove that the properly scaled local time process of WRRW converges in probability to a deterministic function. Using this result we also prove a limit theorem for the position of the random walker at late times.