The Electrical Distribution Feed Box (DFBS) is a 4.5 K saturated liquid helium cryostat constructed for the Large Hadron Collider (LHC) Prototype Cell (String 2). The thermal design of the DFBS is presented, with emphasis on the modelling of the cooling of the current lead chimneys via the helium bath boil-off gas and on the design of the lambda plate. The expected performance is compared to measurements done during the first operation phase of the LHC prototype cell