Let \(N\subset M\) be von Neumann algebras with the same unit element. Denote by \(E(M,N)\) the space of faithful normal conditional expectations \(M\to N\). If the centralizer of \(E\in E(M,N)\) can be identified with the relative commutant \(N^ c\) of \(N\) then \(E\) is called “bécarre” (toned down?). M. Baillet, Y. Denizeau and J.-F. Havet [Compos. Math. 66, No. 2, 199-236 (1988; Zbl 0657.46041)] extended the V. Jones notion of finite index to conditional expectations for pairs of arbitrary von Neumann algebras, extending H. Kosaki’s definition [J. Funct. Anal. 66, 123-140 (1986; Zbl 0607.46034)] of \(E\in E(M,N)\) of being of finite index \([E]\) if \([E]^{-1}=\lambda_ \infty(E)=\max\{\lambda\geq 0\): \(E-\lambda\text{ id}_ M\) is completely positive\(\}>0\), and defining, after M. Pimsner and S. Popa [Ann. Sci. Éc. Norm. Sup., IV. Ser. 19, 57-106 (1986; Zbl 0646.46057)] \(E\) to be strongly of finite index if \(M\), as a right \(N\)-module, has a finite basis orthonormal with respect to \(E\), and weakly of finite index \([E]_ w\) if \([E]_ w^{-1}=\lambda(E)=\max\{\lambda\geq0\): \(E- \lambda\text{ id}_ M\) is positive\(\}\). The author proves that if \(E(M,N)\) contains an element of finite index then there exists a conditional expectation which is bécarre and of minimal index in \(E(M,N)\). For what follows the author assumes that \(M\) has a countable number of orthogonal projections, \(N\) is finite and \(E(M,N)\) non-empty. The author, in [Pac. J. Math. 146, No. 1, 43-70 (1990; Zbl 0663.46052)] has called \(N\) of finite index in \(M\) if there exists a finite representation (as defined there) of the pair \((M,N)\) in a Hilbert space, and the index \([M:N]\) is defined to be the spectral radius of a canonical operator on \(Z(M)\), the centre of \(M\). If some \(E\in E(M,N)\) is strongly of finite index then \(M\) is finite and \(N\) has finite index in \(M\). Conversely, if \(M\) is finite and \(N\) has finite index in \(M\), the three levels of finiteness of index of \(E(M,N)\) are equivalent and the author finds a family of conditional expectations each of finite index. If also \(N^ c\subset N\), there is an \(E_ N\in E(M,N)\), of finite index, canonically associated with the standard trace, such that \([E_ N]=[M:N]\). If \(M\) is finite, either \(Z(M)\) or \(Z(N)\) is atomic, and \(E\) is a conditional expectation weakly of finite index, then \([E]\leq[E]_ w^ 2\). A corollary of this is that if \(M\) is a II\(_ 1\) factor and \(Z(N)\) is finite then every conditional expectation is strongly of finite index and \([E_ N]=[M:N]\).