  The origin of non-baryonic Dark Matter remains elusive despite ongoing
sensitive searches for heavy, thermally produced dark matter particles.
Recently, it has been shown, that non-thermally produced vector bosons
(sometimes called hidden photons) related to a broken U(1) gauge symmetry are
among the possible WISP (weakly interacting slim particles) dark matter
candidates. The WISP Dark Matter eXperiment (WISPDMX) is the first direct
hidden photon dark matter search experiment probing the particle masses within
the 0.8-2.07 $\mu$eV range with four resonant modes of a tunable radio
frequency cavity and down to 0.4 $n$eV outside of resonance. In this paper, we
present the results from the first science run of WISPDMX comprising 22000
broadband spectra with a 500 MHz bandwidth and a 50 Hz spectral resolution,
obtained during 10-second integrations made at each individual tuning step of
the measurements. No plausible dark matter candidate signal is found, both in
the individual spectra reaching minimum detectable power of $8^{-19}$ Watt and
in the averaged spectrum of all the measurements with the minimum detectable
power of $5^{-22}$ Watt attained for a total of 61 \hours of data taking. Using
these spectra, we derive upper limits on the coupling constant of the hidden
photon at the levels of $10^{e-13}$ for the resonant frequency ranges and
$10^{-12}$ for broadband mass range 0.2-2.07 $\mu$eV, and steadily increasing
at masses below 0.2$\mu$eV.
