Step 1: "Magnetization becomes constant" means the material has reached magnetic saturation. At saturation every atomic dipole is already aligned with the field, so the magnetization equals its maximum value \(M_s = n\,m\), where \(n\) is the number density of dipoles and \(m\) is each dipole moment.
Step 2: The Curie law \(M \propto B/T\) applies only in the weak-field, unsaturated regime. Once all dipoles are aligned, the magnetization can no longer grow, so the Curie relation no longer governs it.
Step 3: Lowering the temperature reduces thermal agitation, but the dipoles are already perfectly aligned; there is nothing left to align. Since \(n\) and \(m\) do not change with temperature, the saturation magnetization is unchanged.
\[\boxed{\text{Magnetization remains constant}}\]