Recently, the theory of currents and the existence theory for Plateau's problem have been extended to the case of finite-dimensional currents in infinite-dimensional manifolds or even metric spaces; see [Acta Math. 185 (2000), 1–80] (and also [Proc. Lond. Math. Soc. (3) 106 (2013), 1121–1142], [Adv. Calc. Var. 7 (2014), 227–240] for the most recent developments). In this paper, in the case when the ambient space is Hilbert, we provide the first partial regularity result, in a dense open set of the support, for n-dimensional integral currents which locally minimize the mass. Our proof follows with minor variants [Indiana Univ. Math. J. 31 (1982), 415–434], implementing Lipschitz approximation and harmonic approximation without indirect arguments and with estimates which depend only on the dimension n and not on codimension or dimension of the target space.