An integral domain
R
R
is a finite factorization domain if each nonzero element of
R
R
has only finitely many divisors, up to associates. We show that a Noetherian domain
R
R
is an FFD
⇔
\Leftrightarrow
for each overring
R
′
R’
of
R
R
that is a finitely generated
R
R
-module,
U
(
R
′
)
/
U
(
R
)
U(R’)/U(R)
is finite. For
R
R
local this is also equivalent to each
R
/
[
R
:
R
′
]
R/[R:R’]
being finite. We show that a one-dimensional local domain
(
R
,
M
)
(R,M)
is an FFD
⇔
\Leftrightarrow
either
R
/
M
R/M
is finite or
R
R
is a DVR.