  A method to calculate the nuclear double beta decay ($2\nu\beta\beta$- and
$0\nu\beta\beta$-) amplitudes within the continuum random phase approximation
(cQRPA) is formulated. Calculations of the $\beta\beta$ transition amplitudes
within the cQRPA are performed for ^{76}Ge, ^{100}Mo and ^{130}Te. A rather
simple nuclear Hamiltonian consisting of phenomenological mean field and
zero-range residual particle-hole and particle-particle interaction is used.
The calculated M^{2\nu} are almost not affected when the single-particle
continuum is taken into account. At the same time, a regular suppression of the
$0\nu\beta\beta$-amplitude is found that can be associated with additional
ground state correlations due to collective states in the continuum. It is
expected that future inclusion of the nucleon pairing in the single-particle
continuum will somewhat compensate the suppression.
