  A measurement of the inclusive cross section for the deep-inelastic
scattering of positrons off protons at HERA is presented at momentum transfers
$8.5 \leq Q^2 \leq 35 GeV^2$ and large inelasticity $y = 0.7$, i.e. for the
Bjorken-x range $0.00013 \leq x \leq 0.00055$. Using a next-to-leading order
QCD fit to the structure function F_2 at lower y values, the contribution of
F_2 to the measured cross section at high y is calculated and, by subtraction,
the longitudinal structure function F_{L} is determined for the first time with
an average value of $F_L=0.52 \pm 0.03 (stat)$^ {+0.25}_{-0.22}$ (syst) at
$Q^2=15.4 GeV^2$ and $x=0.000243$.
