  Conway and Coxeter have shown that frieze patterns over positive rational
integers are in bijection with triangulations of polygons. An investigation of
frieze patterns over other subsets of the complex numbers has recently been
initiated by Jorgensen and the first two authors. In this paper we first show
that a ring of algebraic numbers has finitely many units if and only if it is
an order in a quadratic number field $\mathbb{Q}(\sqrt{d})$ where $d<0$. We
conclude that these are exactly the rings of algebraic numbers over which there
are finitely many non-zero frieze patterns for any given height. We then show
that apart from the cases $d\in \{-1,-2,-3,-7,-11\}$ all non-zero frieze
patterns over the rings of integers $\mathcal{O}_d$ for $d<0$ have only
integral entries and hence are known as (twisted) Conway-Coxeter frieze
patterns.
