  EuPd$_2$Si$_2$ is a valence-fluctuating system undergoing a
temperature-induced valence crossover at $T'_V\approx160\,$K. We present the
successful single crystal growth using the Czochralski method for the
substitution series EuPd$_2$(Si$_{1-x}$Ge$_x$)$_2$, with substitution levels
$x\leq 0.15$. A careful determination of the germanium content revealed that
only half of the nominal concentration is build into the crystal structure.
From thermodynamic measurements it is established that $T'_V$ is strongly
suppressed for small substitution levels and antiferromagnetic order from
stable divalent europium emerges for $x\gtrsim 0.10$. The valence transition is
accompanied by a pronounced change of the lattice parameter $a$ of order 1.8%.
In the antiferromagnetically ordered state below $T_N = 47$ K, we find sizeable
magnetic anisotropy with an easy plane perpendicular to the crystallographic c
direction. An entropy analysis revealed that no valence fluctuations are
present for the magnetically ordered materials. Combining the obtained
thermodynamic and structural data, we construct a concentration-temperature
phase diagram demonstrating a rather abrupt change from a valence-fluctuating
to a magnetically-ordered state in EuPd$_2$(Si$_{1-x}$Ge$_x$)$_2$.
