The paper studies the behavior of the modulus \(M_ n(a)\) of the Grötzsch extremal ring \(R_{G,n}(a)\subset {\mathfrak R}^ n\) as a tends to 0. In the first two parts lower and upper estimates are obtained for \[ \lim_{a\to 0}(M_ n(a)+\log a). \] The lower estimates are given in terms of n; the upper estimates are obtained numerically for \(3\leq n\leq 22\); the estimation is based on earlier results of \textit{G. D. Anderson} [Ann. Acad. Sci. Fenn., Ser. A I 575, 1-21 (1974; Zbl 0298.31007); Lect. Notes Math. 743, 10-34 (1979; Zbl 0417.31006)]. The third part provides an upper estimate for \(M_ n(a)/M_ 2(a)\) of correct order as a tends to 0. In the last part lower and upper bounds are given for \(M_ n(a)\) which are of correct order as a tends either to 0 or 1. The paper ends with the suggestion: ``Now we may not be far from the truth about the modulus of the Grötzsch ring.''