  A search for single top quark production is performed in e^\pm p collisions
at HERA. The search exploits data corresponding to an integrated luminosity of
118.3 pb^-1. A model for the anomalous production of top quarks in a flavour
changing neutral current process involving a t-u-gamma coupling is
investigated. Decays of top quarks into a b quark and a W boson are considered
in the leptonic and the hadronic decay channels of the W. Both a cut-based
analysis and a multivariate likelihood analysis are performed to discriminate
anomalous top quark production from Standard Model background processes. In the
leptonic channel, 5 events are found while 1.31 \pm 0.22 events are expected
from the Standard Model background. In the hadronic channel, no excess above
the expectation for Standard Model processes is found. These observations lead
to a cross section \sigma (ep -> e t X) = 0.29 +0.15 -0.14 pb at \sqrt{s} = 319
GeV. Alternatively, assuming that the observed events are due to a statistical
fluctuation, upper limits of 0.55 pb on the anomalous top production cross
section and of 0.27 on the t-u-gamma coupling \kappa_{t-u-gamma} are
established at the 95% confidence level.
