  We use the first 25% of the DEEP2 Galaxy Redshift Survey spectroscopic data
to identify groups and clusters of galaxies in redshift space. The data set
contains 8370 galaxies with confirmed redshifts in the range 0.7 <= z <= 1.4,
over one square degree on the sky. Groups are identified using an algorithm
(the Voronoi-Delaunay Method) that has been shown to accurately reproduce the
statistics of groups in simulated DEEP2-like samples. We optimize this
algorithm for the DEEP2 survey by applying it to realistic mock galaxy catalogs
and assessing the results using a stringent set of criteria for measuring
group-finding success, which we develop and describe in detail here. We find in
particular that the group-finder can successfully identify ~78% of real groups
and that ~79% of the galaxies that are true members of groups can be identified
as such. Conversely, we estimate that ~55% of the groups we find can be
definitively identified with real groups and that ~46% of the galaxies we place
into groups are interloper field galaxies. Most importantly, we find that it is
possible to measure the distribution of groups in redshift and velocity
dispersion, n(sigma,z), to an accuracy limited by cosmic variance, for
dispersions greater than 350km/s. We anticipate that such measurements will
allow strong constraints to be placed on the equation of state of the dark
energy in the future. Finally, we present the first DEEP2 group catalog, which
assigns 32% of the galaxies to 899 distinct groups with two or more members,
153 of which have velocity dispersions above 350km/s. We provide locations,
redshifts and properties for this high-dispersion subsample. This catalog
represents the largest sample to date of spectroscopically detected groups at
z~1.
