In this paper, a 'three block' subspace method for the identification of deterministic bilinear systems is developed. The input signal to the system does not have to be white, which is a major advantage over an existing subspace method for bilinear systems. It is shown that our algorithm provides asymptotically unbiased estimates and the rate at which the bias decreases can be related to a certain data-dependent eigenvalue. Simulation results also show that the new algorithm converges much more rapidly (with sample size) than the existing method. These advantages are achieved by a different arrangement of the input-output equations into 'blocks', and projections onto different spaces than the ones used in the existing method. A further advantage of our algorithm is that the dimensions of the matrices involved are significantly smaller, so that the computational complexity is lower.