Let \(A\) be a Dedekind ring of arithmetic type, and consider a normal, non-central subgroup \(N\) of the special linear group \(\text{SL}_ 2 (A)\). Let \({\mathfrak a}\) be the largest \(A\)-ideal with the property that the elementary matrices \(I_ 2+ aE_{12}\in N\), for all \(a\in {\mathfrak a}\). When \({\mathfrak a}= \{0\}\), \(N\) is said to be a normal subgroup of level zero. In the case of a Dedekind ring \(A\) of arithmetic type with infinitely many units it is known that there are no normal subgroups of level zero. The case of such a ring \(A\) with only finitely many units splits up into the cases: (i) \(A= \mathbb{Z}\), (ii) \(A= {\mathcal O}\) the ring of integers of an imaginary quadratic number field, (iii) \(A\) the coordinate ring of the affine curve obtained by removing a closed point from a projective curve over a finite field. It is shown in this paper that the set of normal subgroups of level zero of the Bianchi groups \(\text{SL}_ 2 ({\mathcal O})\) (i.e. case (ii) above) is uncountable, more precisely, its cardinality is given as \(2^{\aleph_ 0}\). As previously shown, the same result is true in cases (i) and (iii). The proof is based on a result of Zimmert and its generalization [cf. \textit{R. Zimmert}, Invent. Math. 19, 73-81 (1973; Zbl 0254.10019); and \textit{F. Grunewald} and the reviewer, J. Algebra 69, 298-304 (1981; Zbl 0461.20026)].