Let $A$ be a uniform algebra on a compact Hausdorff space $X$ and $m$ a probability measure on $X$. Let $H^p(m)$ be the norm closure of $A$ in $L^p(m)$ with $1 \le p < \infty$ and $H^\infty(m)$ the weak $\ast$ closure of $A$ in $L^\infty(m)$. In this paper, we describe a closed ideal of $A$ and exhibit a closed invariant subspace of $H^p(m)$ for $A$ that is of finite codimension.