We have measured the equilibrium diameter of cylindrical poly(N-isopropylacrylamide) (NIPA) gels with submillimeter diameters under a mechanical constraint along the uniaxial direction. The linear swelling curve, diameter vs temperature, was obtained in the vicinity of the volume phase transition temperature under isometric constraint: both ends of the gel were fixed at rest (stress was zero) in a swollen state at 30 degrees C (a few degrees below the transition temperature). Thereafter, the temperature was gradually changed by keeping the uniaxial length constant (to the fixed length). It was demonstrated that the gel could take a coexistent state at the transition point and remain stable for several days. This time was much longer than the characteristic relaxation time of the phase transition in this tiny gel. The phase coexistence observed here was caused by stress inhomogeneity along the uniaxial direction due to the mechanical constraint. A collapsed to swollen phase transition induced by uniaxial stress at a fixed temperature was also presented. It was shown that the ratio of the swollen portion to the total length could be controlled by the degree of elongation: when the elongation was slightly increased or decreased, the phase boundary between the swollen and collapsed phases was accordingly shifted to increase or decrease the swollen state, respectively. The ratio of the swollen to the collapsed phase in the case of stress-induced coexistence is discussed herein in terms of a phase diagram (diameter vs elongation), and a simple phase selection rule is presented.