  Searches for a stochastic gravitational-wave background (SGWB) using
terrestrial detectors typically involve cross-correlating data from pairs of
detectors. The sensitivity of such cross-correlation analyses depends, among
other things, on the separation between the two detectors: the smaller the
separation, the better the sensitivity. Hence, a co-located detector pair is
more sensitive to a gravitational-wave background than a non-co-located
detector pair. However, co-located detectors are also expected to suffer from
correlated noise from instrumental and environmental effects that could
contaminate the measurement of the background. Hence, methods to identify and
mitigate the effects of correlated noise are necessary to achieve the potential
increase in sensitivity of co-located detectors. Here we report on the first
SGWB analysis using the two LIGO Hanford detectors and address the
complications arising from correlated environmental noise. We apply correlated
noise identification and mitigation techniques to data taken by the two LIGO
Hanford detectors, H1 and H2, during LIGO's fifth science run. At low
frequencies, 40 - 460 Hz, we are unable to sufficiently mitigate the correlated
noise to a level where we may confidently measure or bound the stochastic
gravitational-wave signal. However, at high frequencies, 460-1000 Hz, these
techniques are sufficient to set a $95%$ confidence level (C.L.) upper limit on
the gravitational-wave energy density of \Omega(f)<7.7 x 10^{-4} (f/ 900 Hz)^3,
which improves on the previous upper limit by a factor of $\sim 180$. In doing
so, we demonstrate techniques that will be useful for future searches using
advanced detectors, where correlated noise (e.g., from global magnetic fields)
may affect even widely separated detectors.
