Deep-Inelastic Inclusive ep Scattering at Low x and a Determination of $\alpha_{s}$

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