The authors continue their investigation into the number of topologically different subcontinua of \({\mathbb H}^*\), the Čech-Stone remainder of the half line \({\mathbb H}\). So far in ZFC only nine different subcontinua of \({\mathbb H}^*\) have been discovered. The purpose of this paper is to construct in ZFC a tenth subcontinuum of \({\mathbb H}^*\) and show how to use this continuum to raise the number of different subcontinua of \({\mathbb H}^*\) to fourteen. The authors prove that with the aid of certain subsets of \(\omega^*\) one can parametrize the indecomposable subcontinua of \({\mathbb H}^*\). Then this parametrization is used with a minimal closed and shift-invariant subset of \(\omega^*\) as input to construct the new continuum.