Freezing of a Lennard-Jones fluid between solid surfaces was studied using grand canonical Monte Carlo and molecular dynamics simulations. We explored the formation of frozen phases of hexagonal and orthorhombic symmetry in mono-, bi-, and tri-layer structures. The freezing transition, the type of lattice, and translational and orientational ordering were identified on the basis of orientational order parameters, in-plane two-body and three-body translational correlation functions, orientational correlation functions, and analysis of molecular mobilities. We have found that the freezing temperature is a nonmonotonous function of the pore width: orthorhombic bi-layer freezes at lower temperatures than hexagonal monolayer and hexagonal bi-layer. As the pore width increases, the transition from a hexagonal monolayer to an orthorhombic bi-layer occurred via disordered liquidlike and quasi-long-range four-fold ordered bi-layers. The latter, "quadratic" structure is characterized by an algebraically decaying four-fold orientational correlation function. The transition from crystalline hexagonal bi-layer to orthorhombic tri-layer occurs through a bi-layer structure with two uncoupled hexagonal monolayers. The quadratic phase was observed also as an intermediate structure during freezing of a liquidlike bi-layer into an orthorhombic quasi-crystal. The formation of the quadratic phase was associated with step-wise densification of fluid, a sharp increase of the local orientational order parameter, and a significant reduction of fluid mobility. In the process of solidification, the system passed through a sequence of high-density jammed structures, in which the four-fold symmetry developed progressively, as the temperature decreased. (C) 2003 American Institute of Physics.