  We report heat capacity measurements of SrCu$_2$(BO$_3$)$_2$ under high
pressure along with simulations of relevant quantum spin models and map out the
$(P,T)$ phase diagram of the material. We find a first-order quantum phase
transition between the low-pressure quantum dimer paramagnet and a phase with
signatures of a plaquette-singlet state below T = $2$ K. At higher pressures,
we observe a transition into a previously unknown antiferromagnetic state below
$4$ K. Our findings can be explained within the two-dimensional
Shastry-Sutherland quantum spin model supplemented by weak inter-layer
couplings. The possibility to tune SrCu$_2$(BO$_3$)$_2$ between the
plaquette-singlet and antiferromagnetic states opens opportunities for
experimental tests of quantum field theories and lattice models involving
fractionalized excitations, emergent symmetries, and gauge fluctuations.
