Shape coexistence in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>Ni</mml:mi><mml:mprescripts/><mml:none/><mml:mn>68</mml:mn></mml:mmultiscripts></mml:math>

The internal-conversion and internal-pair-production decays of the first excited 0${}^{+}$ state in $^{68}\mathrm{Ni}$ are studied following the $\ensuremath{\beta}$ decay of $^{68}\mathrm{Co}$. A novel experimental technique, in which the ions of $^{68}\mathrm{Co}$ were implanted into a planar germanium double-sided strip detector and which required digital pulse processing, is developed. The values for the energy of the first excited 0${}^{+}$ state and the electric monopole transition strength from the first excited 0${}^{+}$ state to the ground state in $^{68}\mathrm{Ni}$ are determined to be 1605(3) keV and $7.6(4)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, respectively. Comparisons of the experimental results to Monte Carlo shell-model calculations suggest the coexistence between a spherical ground state and an oblate first excited 0${}^{+}$ state in $^{68}\mathrm{Ni}$.