We present a method for measuring the Hubble parameter , H ( z ) , and angular diameter distance , D _ { A } ( z ) , from the two-dimensional two-point correlation function , and validate it using LasDamas mock galaxy catalogs . Applying our method to the sample of luminous red galaxies ( LRGs ) from the Sloan Digital Sky Survey ( SDSS ) Data Release 7 ( DR7 ) , we measure H ( z = 0.35 ) \equiv H ( 0.35 ) = 82.1 _ { -4.9 } ^ { +4.8 } { km } { s } ^ { -1 } { Mpc } ^ { -1 } , D _ { A } ( z = 0.35 ) \equiv D _ { A } ( 0.35 ) = 1048 _ { -58 } ^ { +60 } Mpc without assuming a dark energy model or a flat Universe . We find that the derived measurements of H ( 0.35 ) r _ { s } ( z _ { d } ) / c and D _ { A } ( 0.35 ) / r _ { s } ( z _ { d } ) ( where r _ { s } ( z _ { d } ) is the sound horizon at the drag epoch ) are nearly uncorrelated , have tighter constraints and are more robust with respect to possible systematic effects . Our galaxy clustering measurements of \ { H ( 0.35 ) r _ { s } ( z _ { d } ) / c,D _ { A } ( 0.35 ) / r _ { s } ( z _ { d } ) \ } = \ { 0.0434 \pm 0.0018 , 6.60 % \pm 0.26 \ } ( with the correlation coefficient r = 0.0604 ) can be used to combine with cosmic microwave background and any other cosmological data sets to constrain dark energy . Our results represent the first measurements of H ( z ) and D _ { A } ( z ) ( or H ( z ) r _ { s } ( z _ { d } ) / c and D _ { A } ( 0.35 ) / r _ { s } ( z _ { d } ) ) from galaxy clustering data . Our work has significant implications for future surveys in establishing the feasibility of measuring both H ( z ) and D _ { A } ( z ) from galaxy clustering data .