For an Ockham algebra \((\mathcal{L}; f)\), the authors consider the ideals \(I\) such that \(I=0/\upsilon\) for some congruence \(\upsilon\) on \((\mathcal{L}; f)\). These ideals are called \textit{congruence kernels} and the congruence \(\upsilon\) for which \(I=0/\upsilon\) is said to have kernel \(I\). The set \(\mathcal{I}_K(\mathcal{L})\) of congruence kernels of \(\mathcal{L}\) ordered by set inclusion is a complete lattice. The authors introduce the set \(\mathcal{I}_2(\mathcal{L}):=\{I\in \mathcal{\mathcal{I}}(L): f^{2}(I)\subseteq I\}\), \(\mathcal{I}(\mathcal{L})\) being the set of ideals of \(\mathcal{L}\). It happens that \(\mathcal{I}_2(\mathcal{L})\) is a complete sub-lattice of \(\mathcal{I}(\mathcal{L})\) and \(\mathcal{I}_K(\mathcal{L})\subseteq \mathcal{I}_2(\mathcal{L})\). For a given ideal \(I\in \mathcal{I}(\mathcal{L})\), the authors define \(\tilde{I}:=\{x\in L: x\wedge f(i)\in I \text{ for some } i\in I\}\) in order to show that \(I\in \mathcal{I}_K(\mathcal{L})\) iff \(f^{2}(I)\subseteq I\) and \(I= \tilde{I}\). For \(I\in \mathcal{I}_K(\mathcal{L})\), the authors show that the smallest congruence with kernel \(I\), call it \(\varphi (I)\), is defined by \[ (x,y)\in \varphi (I) \text{ iff } (\exists i\in I) \text{ such that } (x\vee i)\wedge f(i)= (y\vee i)\wedge f(i). \] To give a description of the largest congruence having a given kernel \(I\in \mathcal{I}_K(\mathcal{L})\), the authors define for each \(a\in L\) and each \(n\in \mathbb{N}\) the set \(W_{a,n}^{I}:= \{x\in L: f^{n}(a)\wedge x\in I\}\) and the binary relation \(\Theta_{n}^{I}:=\{(a,b) : W_{a,n}^{I}=W_{b,n}^{I}\}\) and show that \(\Theta^{I}:=\bigcap_{n\in \mathbb{N}}\Theta_{n}^{I}\) is the largest congruence on \(\mathcal{L}\) having kernel \(I\). The authors give the following characterization of the proper congruence kernels in terms of the prime ideals of \(\mathcal{L}\): \[ L\neq I\in \mathcal{I}_K(\mathcal{L}) \text{ iff } I=\bigcap\{P\in \mathcal{I}_p(\mathcal{L}): I\subseteq P, P\cap f(I)=\emptyset, f^{2}(I)\subseteq P\}. \] \(I_p(L)\) is the set of prime ideals of \(\mathcal{L}\). For a subset \(X\) of \(L\), let \(\kappa (X)\) be the smallest congruence kernel containing \(X\). Write \(\kappa (a)\) when \(X=\{a\}\). The authors show that \(\mathcal{I}_K(\mathcal{L})\) is modular iff for all \(a,b,c\in L\), \(\kappa (a\vee c)\cap \kappa (a\vee b) \subseteq \kappa((\kappa (a\vee c)\cap \kappa (b))\cup \kappa (c))\) and \(\mathcal{I}_K(\mathcal{L})\) is distributive iff for all \(a,b,c\in L\), \(\kappa (a)\cap \kappa (b\vee c)\subseteq \kappa((\kappa(a)\cap \kappa(b))\cup (\kappa(a)\cap \kappa(c)))\).