  Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over
$\mathbb{C}$. A $\operatorname{Y}(\mathfrak{g})$-module is said to be weight if
it is a weight $\mathfrak{g}$-module. We give a complete classification of
simple weight modules for $\operatorname{Y}(\mathfrak{sl}_2)$ which admits a
one-dimensional weight space. We prove that there are four classes of such
modules: finite, highest weight, lowest weight and dense modules. Different
from the classical $\mathfrak{sl}_{2}$ representation theory, we show that
there exist a class of $\operatorname{Y}(\mathfrak{sl}_{2})$ irreducible
modules which have uniformly 2-dimensional weight spaces.
