  One of the features of the unconventional $s_\pm$ state in iron-based
superconductors is possibility to transform to the $s_{++}$ state with the
increase of the nonmagnetic disorder. Detection of such a transition would
prove the existence of the $s_\pm$ state. Here we study the temperature
dependence of the London magnetic penetration depth within the two-band model
for the $s_\pm$ and $s_{++}$ superconductors. By solving Eliashberg equations
accounting for the spin-fluctuation mediated pairing and nonmagnetic impurities
in the $T$-matrix approximation, we have derived a set of specific signatures
of the $s_\pm \to s_{++}$ transition: (1) sharp change in the behavior of the
penetration depth $\lambda_{L}$ as a function of the impurity scattering rate
at low temperatures; (2) before the transition, the slope of $\Delta
\lambda_{L}(T) = \lambda_{L}(T)-\lambda_{L}(0)$ increases as a function of
temperature, and after the transition this value decreases; (3) the sharp jump
in the inverse square of the penetration depth as a function of the impurity
scattering rate, $\lambda_{L}^{-2}(\Gamma_a)$, at the transition; (4) change
from the single-gap behavior in the vicinity of the transition to the two-gap
behavior upon increase of the impurity scattering rate in the superfluid
density $\rho_{s}(T)$.
