The precise determination of the beta-neutrino correlation in the decay of radioactive nuclei is an important tool in the effort to probe fundamental interactions for possible deviation from the Standard Model of particles and fields. In order to realize this, a novel scheme for precision measurement of such correlation in the decay of radioactive helium-6 (half-life 0.8 s) using an electrostatic ion beam trap (EIBT) is being constructed at the Weizmann Institute. Efficient ionization of the expected weak flux of helium-6, together with a bunching characteristic suitable for injection into the EIBT, is thereby essential. Both of these requirements are optimally met by an electron beam ion source/trap (EBIT). In this contribution we present the design of a modified DREEBIT (Dresden EBIT) featuring a direct, nearly closed gas intake into the drift tube assembly for maximum trapping and ionization efficiency for low flux rates, without diminishing the bunching phase space of the EBIT. First tests already indicate significant ion production at a minimal gas flow rate observable by a marginal pressure increase that is close to the detection limit of a Leybold extractor gauge located in close proximity to the drift tube assembly. This is a necessary precondition in order to exploit even tiny amounts of rare isotopes.