  Mixed-order phase transitions display a discontinuity in the order parameter
like first-order transitions yet feature critical behavior like second-order
transitions. Such transitions have been predicted for a broad range of
equilibrium and nonequilibrium systems, but their experimental observation has
remained elusive. Here, we analytically predict and experimentally realize a
mixed-order equilibrium phase transition. Specifically, a discontinuous
solid-solid transition in a two-dimensional crystal of paramagnetic colloidal
particles is induced by a magnetic field $H$. At the transition field
$H_{\text{s}}$, the energy landscape of the system becomes completely flat,
which causes diverging fluctuations and correlation length $\xi\propto | H^2 -
H_{\text{s}}^2|^{-1/2}$. Mean-field critical exponents are predicted, since the
upper critical dimension of the transition is $d_{\text{u}}=2$. Our colloidal
system provides an experimental test bed to probe the unconventional properties
of mixed-order phase transitions.
