  A search for double electron capture ($2\varepsilon$), electron capture with
positron emission ($\varepsilon\beta^+$), and double positron emission
$2\beta^+$) in $^{136}$Ce and $^{138}$Ce was realized with a 465 cm$^3$
ultra-low background HP Ge $\gamma$ spectrometer over 2299 h at the Gran Sasso
underground laboratory. A 627 g sample of cerium oxide deeply purified by
liquid-liquid extraction method was used as a source of $\gamma$ quanta
expected in double $\beta$ decay of the cerium isotopes. New improved half-life
limits were set on different modes and channels of double $\beta$ decay of
$^{136}$Ce and $^{138}$Ce at the level of $T_{1/2}>10^{17}-10^{18}$ yr.

Nuclear Experiment