In this paper a finite-dimensional model is considered which approximates the one-dimensional diffusion process with pointwise observation and controls. The structural properties of the model are investigated, and sufficient conditions for its global identifiability are stated. These involve the input and output matrices of the model and are related to the location of observation and control points. Then the problem of identifying the parameters of the model from the inputs-output data of the distributed system is investigated and the uniqueness of the estimate is discussod. Finally numerical applications are reported. Though the approach is deterministic, the effect of random output noise is tested in the numerical applications.