We analyze the percolation properties of certain clusters defined on configurations of the two-dimensional Heisenberg model. We find that, given any direction in $\stackrel{\ensuremath{\rightarrow}}{n}$ in O(3) space, spins that are almost perpendicular to $\stackrel{\ensuremath{\rightarrow}}{n}$ form a percolating cluster. This result indicates how the model can avoid a previously conjectured Kosterlitz-Thouless phase transition at a finite temperature T.