A chain order of a skew field
D
D
is a subring
R
R
of
D
D
so that
d
∈
D
∖
R
d\in D\backslash R
implies
d
−
1
∈
R
.
d^{-1}\in R.
Such a ring
R
R
has rank one if
J
(
R
)
J(R)
, the Jacobson radical of
R
,
R,
is its only nonzero completely prime ideal. We show that a rank one chain order of
D
D
is either invariant, in which case
R
R
corresponds to a real-valued valuation of
D
,
D,
or
R
R
is nearly simple, in which case
R
,
R,
J
(
R
)
J(R)
and
(
0
)
(0)
are the only ideals of
R
,
R,
or
R
R
is exceptional in which case
R
R
contains a prime ideal
Q
Q
that is not completely prime. We use the group
M
(
R
)
\mathcal {M}(R)
of divisorial
R
R
-ideals of
D
D
with the subgroup
H
(
R
)
\mathcal {H}(R)
of principal
R
R
-ideals to characterize these cases. The exceptional case subdivides further into infinitely many cases depending on the index
k
k
of
H
(
R
)
\mathcal {H}(R)
in
M
(
R
)
.
\mathcal {M}(R).
Using the covering group
G
\mathbb {G}
of
SL
(
2
,
R
)
\operatorname {SL}(2,\mathbb {R})
and the result that the group ring
T
G
T\mathbb {G}
is embeddable into a skew field for
T
T
a skew field, examples of rank one chain orders are constructed for each possible exceptional case.