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The object of the present study is the integrated density of states of aquantum particle in multi-dimensional Euclidean space which is characterized bya Schr\"odinger operator with a constant magnetic field and a random potentialwhich may be unbounded from above and from below. For an ergodic randompotential satisfying a simple moment condition, we give a detailed proof thatthe infinite-volume limits of spatial eigenvalue concentrations offinite-volume operators with different boundary conditions exist almost surely.Since all these limits are shown to coincide with the expectation of the traceof the spatially localized spectral family of the infinite-volume operator, theintegrated density of states is almost surely non-random and independent of thechosen boundary condition. Our proof of the independence of the boundarycondition builds on and generalizes certain results by S. Doi, A. Iwatsuka andT. Mine [Math. Z. {\bf 237} (2001) 335-371] and S. Nakamura [J. Funct. Anal.{\bf 173} (2001) 136-152].Comment: This paper is a revised version of the first part of the first  version of math-ph/0010013. For a revised version of the second part, see  math-ph/0105046. To appear in Reviews in Mathematical Physic

EXISTENCE AND UNIQUENESS OF THE INTEGRATED DENSITY OF STATES FOR SCHRÖDINGER OPERATORS WITH MAGNETIC FIELDS AND UNBOUNDED RANDOM POTENTIALS