For a domain
D
D
in
R
¯
n
\bar {\mathbb R}^n
, the modulus metric is defined by
μ
D
(
x
,
y
)
=
inf
γ
cap
(
D
,
γ
)
\mu _D(x,y)=\inf _\gamma \textrm {cap}(D,\gamma )
, where the infimum is taken over all curves
γ
\gamma
in
D
D
joining
x
x
to
y
y
, and “cap" denotes the conformal capacity of the condensers. It has been conjectured by J. Ferrand, G. J. Martin, and M. Vuorinen that isometries in the modulus metric are conformal mappings. We prove the conjecture when
n
=
2
n=2
. In higher dimensions, we prove that isometries are quasiconformal mappings.