Detecting a stochastic gravitational wave background requires that we first understand and model any astrophysical foregrounds. In the millihertz frequency band, the predominate foreground signal will be from unresolved white dwarf binaries in the galaxy. We build on our previous work to show that a stochastic gravitational wave background can be detected in the presence of both instrument noise and a galactic confusion foreground. The key to our approach is accurately modeling the spectra for each of the various signal components. We simulate data for a gigameter Laser Interferometer Space Antenna operating in the millihertz frequency band with both six and four links. We obtain posterior distribution functions for the instrument noise parameters, the galaxy level and modulation parameters, and the stochastic background energy density. We find that we are able to detect a scale-invariant stochastic background with energy density as low as ${\mathrm{\Omega }}_{\mathrm{gw}}=2\times 10^{-13}$ for a six-link interferometer and ${\mathrm{\Omega }}_{\mathrm{gw}}=5\times 10^{-13}$ for a four-link interferometer with one year of data.

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