  We present exactly simultaneous X-ray and TeV monitoring with {\it RXTE} and
HEGRA of the TeV blazar Mrk 501 during 15 days in 1998 June. After an initial
period of very low flux at both wavelengths, the source underwent a remarkable
flare in the TeV and X-ray energy bands, lasting for about six days and with a
larger amplitude at TeV energies than in the X-ray band. At the peak of the TeV
flare, rapid TeV flux variability on sub-hour timescales is found. Large
spectral variations are observed at X-rays, with the 3--20 keV photon index of
a pure power law continuum flattening from $\Gamma=2.3$ to $\Gamma=1.8$ on a
timescale of 2--3 days. This implies that during the maximum of the TeV
activity, the synchrotron peak shifted to energies $\gtrsim 50$ keV, a behavior
similar to that observed during the longer-lasting, more intense flare in 1997
April. The TeV spectrum during the flare is described by a power law with
photon index $\Gamma=1.9$ and an exponential cutoff at $\sim$ 4 TeV; an
indication for spectral softening during the flare decay is observed in the TeV
hardness ratios. Our results generally support a scenario where the TeV photons
are emitted via inverse Compton scattering of ambient seed photons by the same
electron population responsible for the synchrotron X-rays. The simultaneous
spectral energy distributions (SEDs) can be fit with a one-zone
synchrotron-self Compton model assuming a substantial increase of the magnetic
field and the electron energy by a factor of 3 and 10, respectively.
