We classify closed curves isomorphic to the affine line in the complement of a smooth rational projective plane conic $Q$. Over a field of characteristic zero, we show that up to the action of the subgroup of the Cremona group of the plane consisting of birational endomorphisms restricting to biregular automorphisms outside $Q$, there are exactly two such lines: the restriction of a smooth conic osculating $Q$ at a rational point and the restriction of the tangent line to $Q$ at a rational point. In contrast, we give examples illustrating the fact that over fields of positive characteristic, there exist exotic closed embeddings of the affine line in the complement of $Q$. We also determine an explicit set of birational endomorphisms of the plane whose restrictions generates the automorphism group of the complement of $Q$ over a field of arbitrary characteristic.