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Summary form only given. We consider the phenomenological model of the nonlocal Kerr-type nonlinear medium. We obtain the nonlinear Schroedinger equation describing propagation of the one-dimensional optical beam in a weakly nonlinear medium. It turns out that this nonlinear equation can be exactly solved for both focusing and defocusing nonlinearity. In this way one can obtain the intensity profile of bright and dark solitons as a function of the nonlocality parameter. An important aspect of any soliton solution is its stability. Using the well known stability criteria developed for 1D solitons, we found that bright as well as dark spatial solitons in weakly nonlocal nonlinear medium are stable. We confirmed this result by numerical simulation of the propagation equation using exact soliton solutions as initial condition. Solitons propagate in stable fashion over a distances of many diffraction lengths. We also studied collisions of these solitons. We found that, for large intersecting angle, the collision is basically elastic. However, for small angle (long interaction) nonlocal solitons collide inelastically, in analogy to solitons of other nonintegrable models. An example of inelastic collision of two identical nonlocal dark solitons is shown.