For a field F of characteristic zero and an additive subgroup G of (F), a Lie algebra (B)(G) of the Block type is defined with the basis {Lα,i, c|α ∈ G, -1 ≤ i ∈ (Z)} and the relations [Lα,i, Lβ,j] =((i + 1)β - (j + 1)α)Lα+β,i+j + αδα,-βδi+j,-2c, [c, Lα,i] = 0.Given a total order (γ) on G compatible with its group structure, and any Λ ∈ (B)(G)*0, a Verma (B)(G)-module M(Λ, (γ)) is defined, and the irreducibility of M(Λ,(γ)) is completely determined. Furthermore, it is proved that an irreducible highest weight (B)((Z))-module is quasifinite if and only if it is a proper quotient of a Verma module.