  In this Letter we present new, genuinely non-Abelian vortex solutions in
SU(2) Yang-Mills--Higgs theory with only one {\it isovector} scalar field.
These non-Abelian solutions branch off their Abelian counterparts
(Abrikosov-Nielsen-Olesen vortices) for precise values of the Higgs potential
coupling constant $\lambda$. For all values of $\lambda$, their energies lie
below those of the Abelian energy profiles, the latter being logarithmically
divergent as $\lambda\to\infty$. The non-Abelian branches plateau in the limit
$\lambda\to\infty$ and their number increases with $\lambda$, this number
becoming infinite. For each vorticity, the gaps between the plateauing energy
levels become constant. In this limit the non-Abelian vortices are
non-interacting and are described by the {\it self-dual} vortices of the O(3)
sigma model. In the absence of a topological lower bound, we expect these
non-Abelian vortices to be {\it sphalerons}.
