  In this paper, a new class of $\Z$-graded Lie conformal algebras $\CW(a,c)$
of infinite rank is constructed.
  The conformal derivations and one-dimensional central extensions of
$\CW(a,c)$ are completely determined. And all conformal modules of rank one
over $\CW(a,c) (a\neq0)$ are proved to be trivial and all such nontrivial
(irreducible) modules over $\CW(0,c)$ are classified.
