We give two characterizations of a one-dimensional Henselian domain. If
(
A
,
M
)
\left ( {A,\mathcal {M}} \right )
is a local domain of Krull dimension at least two, or if
(
A
,
M
)
\left ( {A,\mathcal {M}} \right )
is a one-dimensional Henselian local domain, then a polynomial
f
f
in
A
[
T
]
A\left [ T \right ]
is Weierstrass if and only if
(
M
,
T
)
\left ( {\mathcal {M},T} \right )
is the only maximal ideal of
A
[
T
]
A\left [ T \right ]
that contains
f
f
.