We investigate the relation between the star formation rate surface density (Sigma_SFR) and the mass surface density of gas (Sigma_gas) in NGC 5194. VIRUS-P integral field spectroscopy is used to measure H-alpha, H-beta [NII]6548,6584, and [SII]6717,6731 emission line fluxes for 735 regions ~170 pc in diameter, and derive extinction corrected Sigma_SFRs. HI 21cm and CO maps are used to measure the HI and H_2 gas surface density for each region. We present a new method for fitting the Star Formation Law (SFL), which includes the intrinsic scatter in the relation as a free parameter, allows the inclusion of non-detections, and is free of the systematics involved in performing linear correlations over incomplete data in logarithmic space. We use the [SII]/H-alpha ratio to separate the H-alpha flux contribution from the diffuse ionized gas (DIG). After removing the DIG, we measure a slope N=0.82+/-0.05, and an intrinsic scatter epsilon=0.43+/-0.02 dex for the molecular gas SFL. We also measure a typical depletion timescale tau~2 Gyr, in good agreement with Bigiel et al. (2008). The HI density shows no correlation with the SFR, and the total gas SFL closely follows the molecular gas SFL. We assess the validity of corrections applied in narrow-band H-alpha measurements to overcome issues related to continuum subtraction, underlying photospheric absorption, and contamination by the [NII] doublet. The disagreement with the super-linear molecular SFL measured by Kennicutt et al. (2007) is due to differences in the fitting method. Our results support a low and close to constant star formation efficiency (SFE = 1/tau) in the molecular ISM. The data shows excellent agreement with the model of Krumholz et al (2009). The large intrinsic scatter may imply the existence of other important parameters setting the SFR.Comment: 23 pages, 19 figures, 14 pages of tables. Accepted for publication in ApJ. (Figure 16 has been corrected from the first submitted version.