  In this paper, we introduce a class of infinite Lie conformal algebras
$\mathfrak{B}(\alpha,\beta,p)$, which are the semi-direct sums of Block type
Lie conformal algebra $\mathfrak{B}(p)$ and its non-trivial conformal modules
of $\Z$-graded free intermediate series. The annihilation algebras are a class
of infinite-dimensional Lie algebras, which include a lot of interesting
subalgebras: Virasoro algebra, Block type Lie algebra, twisted
Heisenberg-Virasoro algebra and so on.
  We give a complete classification of all finite non-trivial irreducible
conformal modules of $\mathfrak{B}(\alpha,\beta,p)$ for $\alpha,\beta\in\C,
p\in\C^*$. As an application, the classifications of finite irreducible
conformal modules over a series of finite Lie conformal algebras
$\mathfrak{b}(n)$ for $n\geq1$ are given.
