Summary: In the present paper, we study the following scaled nonlinear Schrödinger equation (NLS) in one space dimension: \[ i\frac{\mathrm {d}}{{\mathrm {d}}t}\psi^{\varepsilon}(t)=-\Delta\psi^{\varepsilon}(t) +\frac{1}{\varepsilon}V\left(\frac{x}{\varepsilon} \right)|\psi^{\varepsilon}(t)|^{2\mu}\psi^{\varepsilon}(t) \quad \varepsilon > 0\quad V\in L^1(\mathbb{R},(1+|x|){\mathrm {d}}x) \cap L^\infty(\mathbb{R}). \] This equation represents a nonlinear Schrödinger equation with a spatially concentrated nonlinearity. We show that in the limit \({\varepsilon\to 0}\) the weak (integral) dynamics converges in \({H^1(\mathbb{R})}\) to the weak dynamics of the NLS with point-concentrated nonlinearity: \[ {i\frac{{\mathrm {d}}}{{\mathrm {d}}t} \psi(t) =H_{\alpha} \psi(t).} \] where \({H_\alpha}\) is the Laplacian with the nonlinear boundary condition at the origin \({\psi'(t,0+)-\psi'(t,0-)=\alpha|\psi(t,0)|^{2\mu}\psi(t,0)}\) and \({\alpha=\int_{\mathbb{R}}V{\mathrm {d}}x}\). The convergence occurs for every \({\mu\in \mathbb{R}^+}\) if \(V \geq~ 0 \) and for every \({\mu\in (0,1)}\) otherwise. The same result holds true for a nonlinearity with an arbitrary number \(N\) of concentration points.