Normal projective varieties and its coordinate ring

Question

Let $k[X_0,...,X_n]$ be a polynomial ring over an algebraically closed field of characteristic zero and $I$ an ideal of $k[X_0,...,X_n]$ generated by homogenous polynomials. Denote by $X$ the projective scheme Proj $k[X_0,...,X_n]/I$. Assume that $k[X_0,...,X_n]/I$ is an integral domain. Is it true that $X$ is a normal scheme if and only if the ring $k[X_0,...,X_n]/I$ is normal?