Central galaxies in galaxy clusters may be key discriminants in the competition between the cold dark matter (CDM) paradigm and modified Newtonian dynamics (MOND). We investigate the dark halo of NGC 1399, the central galaxy of the Fornax cluster, out to a galactocentric distance of 80 kpc. The data base consists of 656 radial velocities of globular clusters obtained with MXU/VLT and GMOS/Gemini, which is the largest sample so far for any galaxy. We performed a Jeans analysis for a non-rotating isotropic model. An NFW halo with the parameters r_s = 50 kpc and rho_s = 0.0065 M_sun/pc^3 provides a good description of our data, fitting well to the X-ray mass. More massive halos are also permitted that agree with the mass of the Fornax cluster as derived from galaxy velocities. We compare this halo with the expected MOND models under isotropy and find that additional dark matter on the order of the stellar mass is needed to get agreement. A fully radial infinite globular cluster system would be needed to change this conclusion. Regarding CDM, we cannot draw firm conclusions. To really constrain a cluster wide halo, more data covering a larger radius are necessary. The MOND result appears as a small-scale variant of the finding that MOND in galaxy clusters still needs dark matter.