  We report the results of a detailed analysis of the temperature structure of
the X-ray emitting plasma halo of M~87, the cD galaxy of the Virgo Cluster.
Using the MEKAL model, the data provide strong indications that the
intracluster medium has a single phase structure locally, except the regions
associated to the radio structures. The deprojected spectrum at each radius is
well fitted by a single temperature MEKAL model, except for the very central
region ($<$ 2 arcmin) which seems to be affected by the jet and radio lobe
structure. The temperature of the intracluster plasma is 1 keV at the center
and gradually increases to 2.5 keV at 80 kpc. We have also fitted spectra using
the APEC code. Although the large changes of the strength of K$\alpha$ lines
causes a discrepancy between the Fe-L and Fe-K lines for the APEC results, the
overall temperature structure has not changed. There is no sign of excess
absorption in the spectral data. The single-phase nature of the intracluster
medium is in conflict with the standard cooling flow model which is based on a
multi-phase temperature structure. In addition, the signature of gas cooling
below 0.8 keV to zero temperature is not observed as expected for a cooling
flow. The gravitational mass profile derived from the temperature and density
distribution of the intracluster gas shows two distinct contributions that can
be assigned to the gravitational potential of the cD galaxy and the cluster.
The central temperature of the intracluster medium agrees well with the
potential depth and the velocity dispersion of the cD galaxy. The latter result
implies that the central region of the intracluster medium is equivalent to a
virialized interstellar medium in M 87.
