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This paper is concerned with the observability problem of a parallel-flow two-fluid heat exchanger equation with diffusion term. First, the case where two fluid temperatures are measured at the outlet is considered. It is shown that the observed system with the measurements becomes observable on any interval of time through a concrete series expression of the solution. Next, the two cases where each one of two fluid temperatures is measured at the outlet are considered. It is also shown that the observed system with the only one measurement becomes observable on any interval of time except for the special cases of physical constants appearing in the equation. For the exceptional cases the unobservable subspace is finite dimensional and is characterized by using the eigenfunctions of heat equation with fluid transfer term.