The attenuation of pressure waves in a saturated bubbly magma is examined in a model, coupling seismic wave-propagation with bubble growth dynamics. This model is solved analytically and numerically, including effects of diffusion of volatiles, visco-elasticity and bubble number density. We show that wave attenuation is controlled mainly by the Peclet and Deborah numbers. The Peclet number is a measure of the relative importance of advection to diffusion. The Deborah number is a visco-elastic measure, describing the importance of elasticity in comparison to viscous melt deformation. We solve numerically for wave attenuation for various magma properties corresponding to a wide range of Peclet and Deborah numbers. We show that the numerical solution can be approximated quite well for frequencies above 1 Hz, by an analytical end-member solution, obtained for high Peclet and low Deborah numbers. For lower frequencies, volatile transport should be accounted for, leading to higher attenuation with respect to the analytical solution. However, if the Deborah number is increased, either by longer relaxation time or by higher frequencies, then attenuation decreases with respect to the analytical solution. Therefore, visco-elasticity leads to a significant improvement of the resonating qualities of a magma-filled conduit and widens the depth and frequency ranges where pressure waves will propagate efficiently through the conduit.