The demixing of a 0.002 wt % solution of a poly(epsilon -caprolactone) chain (M-w = 20 000; M-w/M-n = 1.07), labeled at both chain ends with pyrene, in tetrahydrofuran (THF) was studied using fluorescence. The very diluted region of the phase diagram of small polymer chains was explored for the first time. The coexistence curve was crossed at -30 degreesC, below the coil-globule transition temperature, which was observed at 0 degreesC. The demixing occurred from a "gas" solution of globules and small clusters, and we observed three stages before precipitation: an initial frozen stage, where particles collided but did not grow; a second region where clusters grew by incorporation of globules; and a third stage where coarsening occurred. The excimer-to-monomer fluorescence intensities ratio followed power law type dependencies in time, during both the growth and coarsening stages. This was attributed to phase demixing slowed by viscoelastic effects. The demixing was accelerated by deeper temperature quenches and retarded by slower rates of quenching.