Using the dynamic method to evolve a weighted network, a new weighted network with accelerated attachment of newly added edges is proposed. At each time step, the number of newly added edges is proportional to the size of the network. The degree distribution of the network is discussed. The strength distribution is also obtained, and the analytical relation between the strength and the degrees of nodes is presented. The network shows scale-free property in both theoretical analysis and numerical simulations. In addition, regardless of degree preferential attachment or strength preferential attachment in the evolution process, as long as the weight of edge follows a certain probability distribution, the degree distribution always follows the power-law distribution and its exponent is not related with the attachment mechanism and the specific probability distribution.