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Let D denote a Dirac operator on a compact odd-dimensional manifold M with boundary Y . The elliptic boundary value problem D P is the operator D with domain determined by a boundary condition P from the smooth self-adjoint Grassmannian Gr * ∞ (D). It has a welldefined ζ-determinant (see [Wo5]). The determinant line bundle over Gr * ∞ (D) has a natural trivialization in which the canonical Quillen determinant section becomes a function, denoted by det C D P , equal to the Fredholm determinant of a naturally associated operator on the space of boundary sections. In this paper we show that the ζ-regularized determinant det ζ D P is equal to det C D P modulo a natural multiplicative constant.

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