  We consider the \emph{exact} error correction of a noisy Euclidean distance
matrix, EDM, where the elements are the squared distances between $n$ points in
$R^d$. For our problem we are given two facts: (i) the embedding dimension,
$d$, (ii) \emph{exactly one} distance in the data is corrupted by \emph{nonzero
noise}. But we do \underline{not} know the magnitude nor position of the noise.
Thus there is a combinatorial element to the problem. We present three solution
techniques. These use three divide and conquer strategies in combination with
three versions of facial reduction that use: exposing vectors, facial vectors,
and Gale transforms. This sheds light on the connections between the various
forms of facial reduction related to Gale transforms. Our highly successful
empirics confirm the success of these approaches as we can solve huge problems
of the order of $100,000$ nodes in approximately one minute to machine
precision. \\Our algorithm depends on identifying whether a principal submatrix
of the \EDM contains the corrupted element. We provide a theorem for doing this
that is related to the existing results for identifying \emph{yielding}
elements, i.e.,~we provide a characterization for guaranteeing the perturbed
EDM remains an EDM with embedding dimension $d$. The characterization is
particularly simple in the $d=2$ case. \\In addition, we characterize when the
intuitive approach of the nearest EDM problem, solves our problem. In fact, we
show that this happens if, and only if, the original distance element is $0$,
degenerate, and the perturbation is negative.
