Let $R$ be an affine $k$-domain over the field $k$. The paper’s main result is that if $R$ admits a nontrivial embedding in a polynomial ring $K\left[s\right]$ for some field $K$ containing $k$, then $R$ can be embedded in a polynomial ring $F\left[t\right]$ which extends $R$ algebraically. This theorem can be applied to subrings of a ring which admits a nonzero locally nilpotent derivation. In this way, we obtain a concise new proof of the cancellation theorem for rings of transcendence degree one for fields of characteristic $0$.