If

$A$ is a uniform algebra contained in

$C\left(X\right)$, where

$X$ is a compact Hausdorff space then

$A$ is called hypo-Dirichlet if the uniform closure of

$A+\overline{A}$ is of finite codimension in

$C\left(X\right)$ (if the closure of

$A+\overline{A}=C\left(X\right)$ then

$A$ is Dirichlet). A representation

$f\to {T}_{f}$ of

$A$ on the Hilbert space

$H$ is said to have a

$\rho $-dilation

$\varphi \to {U}_{\varphi}$ (where

$\rho \in {\mathbb{R}}_{+}$) to

$C\left(X\right)$ (and a larger Hilbert space

$K$) if

${T}_{f}=\rho P{U}_{f}|H$, where

$P:K\to H$ is the orthogonal projection. While it is known that only two hypo-Dirichlet (non-Dirichlet) algebras have 1-dilations, it is shown here that if there is a non-zero complex homomorphism

$\tau $ of

$A$ for which the representing measures are given by

$hdm$, where

$m$ is the core measure and

$h$ is continuous on

$X$ (and not just essentially bounded with respect to

$m$), then any representation of

$A$ has a

$\rho $-dilation for some

$\rho $. The argument is based on Naimark’s dilation theorem and on the observation that with the assumption that the space

${N}_{\tau}$ of representing measures for

$\tau $ is a subspace of

$C\left(X\right)$,

$C\left(X\right)$ is spanned by

$A$,

$\overline{A}$, and

${N}_{\tau}$. Several examples, e.g. certain subalgebras of the disc algebra, and corollaries are included.