The main physics programme of the International Linear Collider (ILC) requires a measurement of the beam energy at the interaction point with an accuracy of $10^{-4}$ or better. To achieve this goal a magnetic spectrometer using high resolution beam position monitors (BPMs) has been proposed. This paper reports on the cavity BPM system that was deployed to test this proposal. We demonstrate sub-micron resolution and micron level stability over 20 hours for a $1m$ long BPM triplet. We find micron-level stability over 1 hour for 3 BPM stations distributed over a $30m$ long baseline. The understanding of the behaviour and response of the BPMs gained from this work has allowed full spectrometer tests to be carried out.
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