  Recently, a novel material with bilayer kagome lattice
Ca$_{10}$Cr$_7$O$_{28}$ was proposed to be a gapless quantum spin liquid, due
to the lack of long-range magnetic order and the observation of broad diffuse
excitations. Here, we present the ultralow-temperature thermal conductivity
measurements on single crystals of Ca$_{10}$Cr$_7$O$_{28}$ to detect its
low-lying magnetic excitations. At finite temperatures, with increasing the
magnetic fields, the thermal conductivity exhibits a clear dip around 6 T,
which may correspond to a crossover in the magnetic ground state. At the
zero-temperature limit, no residual linear term is found at any fields,
indicating the absence of gapless itinerant fermionic excitations. Therefore,
if the spinons do exist, they are either localized or gapped. In the gapped
case, the fitting of our data gives a small gap $\Delta \sim$ 0.27(2) K. These
results put strong constraints on the theoretical description of the ground
state in this quantum spin liquid candidate.
