Affine $k$-domain of dimension$1$ can always be embedded in a polynomial ring?

Question

Let $k$ be an algebraically closed field. Let $R$ be a UFD which is a finitely generated$k$-algebra.

If $\dim R=1$, then is it true that there exists an injective $k$-algebra homomorphism from $R$ to $k[X_1,...,X_n]$ for some $n$ ? Can we drop the condition $\dim R=1$ ?

(NOTE: For finitely generated$k$-algebra $R$, which is also an integral domain, $\dim R=tr.deg_k R$ )