The longitudinal spin transfer, $D_{LL}$, from high energy polarized protons to $\Lambda$ and $\bar{\Lambda}$ hyperons has been measured for the first time in proton-proton collisions at $\sqrt{s} = 200 \mathrm{GeV}$ with the STAR detector at RHIC. The measurements cover pseudorapidity, $\eta$, in the range $|\eta| < 1.2$ and transverse momenta, $p_\mathrm{T}$, up to $4 \mathrm{GeV}/c$. The longitudinal spin transfer is found to be $D_{LL}= -0.03\pm 0.13(\mathrm{stat}) \pm 0.04(\mathrm{syst})$ for inclusive $\Lambda$ and $D_{LL} = -0.12 \pm 0.08(\mathrm{stat}) \pm 0.03(\mathrm{syst})$ for inclusive $\bar{\Lambda}$ hyperons with $<\eta> = 0.5$ and $ = 3.7 \mathrm{GeV}/c$. The dependence on $\eta$ and $p_\mathrm{T}$ is presented.