  We investigated the superconducting state of the noncentrosymmetric
superconductors Li$_2$Pd$_x$Pt$_{3-x}$B with superconducting transition
temperature $T_c$= 5.16(8) K ($x$=2.25), 3.56(8) K ($x=1.5$) and 2.60 K ($x=0$)
by means of muon-spin rotation ($\mu$SR) and specific heat experiments. The
$\mu$SR relaxation rate $\sigma_{sc}$ was found to be constant at low
temperatures for all the compounds. Data taken at different magnetic fields
show that the magnetic penetration depth $\lambda$ is field-independent for
Li$_2$Pd$_{2.25}$Pt$_{0.75}$B and Li$_2$Pt$_{3}$B. The electronic contribution
to the specific heat measured in Li$_2$Pd$_{1.5}$Pt$_{1.5}$B and
Li$_2$Pt$_{3}$B increases exponentially at the lowest temperatures. These
features suggest that the {\it whole family} of Li$_2$Pd$_x$Pt$_{3-x}$B are
single-gap s-wave superconductors across the entire doping regime.
