The stiffness of the hadronic equation of state has been extracted from the production rate of $K^+$ mesons in heavy ion collisions around 1 $A$ GeV incident energy. The data are best described with a compressibility coefficient $\kappa$ around 200 MeV, a value which is usually called ``soft''. This is concluded from a detailed comparison of the results of transport theories with the experimental data using two different procedures: (i) the energy dependence of the ratio of $K^+$ from Au+Au and C+C collisions and (ii) the centrality dependence of the $K^+$ multiplicities. It is demonstrated that input quantities of these transport theories which are not precisely known, like the kaon-nucleon potential, the $\Delta N \to N K^+ \Lambda$ cross section or the life time of the $\Delta$ in matter do not modify this conclusion.