We present the results of the first combined dark matter search targeting the Galactic Center using the ANTARES and IceCube neutrino telescopes. For dark matter particles with masses from 50 to 1000 GeV, the sensitivities on the self-annihilation cross section set by ANTARES and IceCube are comparable, making this mass range particularly interesting for a joint analysis. Dark matter self-annihilation through the tau(+)tau(-) , mu(+)mu(-) , b (b) over bar, and W+W- channels is considered for both the Navarro-Frenk-White and Burkert halo profiles. In the combination of 2101.6 days of ANTARES data and 1007 days of IceCube data, no excess over the expected background is observed. Limits on the thermally averaged dark matter annihilation cross section <sigma(A)upsilon > are set. These limits present an improvement of up to a factor of 2 in the studied dark matter mass range with respect to the individual limits published by both collaborations. When considering dark matter particles with a mass of 200 GeV annihilating through the tau(+)tau(-)channel, the value obtained for the limit is 7.44 x 10(-24) cm(3) s(-1 )for the Navarro-Frenk-White halo profile. For the purpose of this joint analysis, the model parameters and the likelihood are unified, providing a benchmark for forthcoming dark matter searches performed by neutrino telescopes.