  A precise measurement of the inclusive deep-inelastic e^+p scattering cross
section is reported in the kinematic range 1.5<= Q^2 <=150 GeV^2 and
3*10^(-5)<= x <=0.2. The data were recorded with the H1 detector at HERA in
1996 and 1997, and correspond to an integrated luminosity of 20 pb^(-1). The
double differential cross section, from which the proton structure function
F_2(x,Q^2) and the longitudinal structure function F_L(x,Q^2) are extracted, is
measured with typically 1% statistical and 3% systematic uncertainties. The
measured partial derivative (dF_2(x,Q^2)/dln Q^2)_x is observed to rise
continuously towards small x for fixed Q^2. The cross section data are combined
with published H1 measurements at high Q^2 for a next-to-leading order DGLAP
QCD analysis.The H1 data determine the gluon momentum distribution in the range
3*10^(-4)<= x <=0.1 to within an experimental accuracy of about 3% for Q^2 =20
GeV^2. A fit of the H1 measurements and the mu p data of the BCDMS
collaboration allows the strong coupling constant alpha_s and the gluon
distribution to be simultaneously determined. A value of alpha
_s(M_Z^2)=0.1150+-0.0017 (exp) +0.0009-0.0005 (model) is obtained in NLO, with
an additional theoretical uncertainty of about +-0.005, mainly due to the
uncertainty of the renormalisation scale.
