  We report measurements of the in-plane London penetration depth $\lambda$ in
single crystals of the $\alpha$-PdBi$_{2}$ superconductor --- the
$\alpha$-phase counterpart of the putative topological superconductor
$\beta$-PdBi$_{2}$, down to 0.35~K using a high-resolution tunnel-diode-based
technique. Both $\lambda$ and superfluid density $\rho_{s}$ exhibit an
exponential behavior for $T\leq$ 0.35$T_{c}$, with
$\Delta(0)/k_{B}T_{c}\sim$2.0, $\Delta C/\gamma T_{c}$$\sim$2.0 and
$\lambda(0)$$\sim$140~nm, showing that $\alpha$-PdBi$_{2}$ is a
moderately-coupling, fully-gapped superconductor. The values of $\Delta(0)$ and
$\Delta C/\gamma T_{c}$ are consistent with each other via strong-coupling
corrections.
