In this work we are going to show weak convergence of a probability measure corresponding to the solution of the following nonlinear stochastic heat equation $\frac{\partial}{\partial t} u_{t}(x) = \fracκ{2} Δu_{ t}(x) + σ(u_{t}(x))η_α$ with colored noise $η_α$ to the measure corresponding to the solution of the same equation but with white noise $η$ as $α\uparrow 1$ on the space of continuous functions with compact support. The noise $η_α$ is assumed to be colored in space and its covariance is given by $\operatorname{E} \left[ η_α(t,x) η_α(s,y) \right] = δ(t-s) f_α(x-y)$ where $f_α$ is the Riesz kernel $f_α(x) \propto 1/\left|x\right|^α$. We will also state a result about continuity of measure in $α$, for $α\in (0,1)$. We will work with the classical notion of weak convergence of measures.