We present a revised measurement of the mass of the central black hole (Mbh) in the Seyfert 1 galaxy NGC 4151. The new stellar dynamical mass measurement is derived by applying an axisymmetric orbit-superposition code to near-infrared integral field data obtained using adaptive optics with the Gemini NIFS spectrograph. When our models attempt to fit both the NIFS kinematics and additional low spatial resolution kinematics, our results depend sensitively on how chi-squared is computed--probably a consequence of complex bar kinematics that manifest immediately outside the nuclear region. The most robust results are obtained when only the high spatial resolution kinematic constraints in the nuclear region are included in the fit. Our best estimates for the BH mass and H-band mass-to-light ratio are Mbh~(3.76+/-1.15)E7 Msun (1-sigma error) and M/L(H-band)~0.34+/-0.03 Msun/Lsun (3-sigma error), respectively (the quoted errors reflect the model uncertainties). Our BH mass measurement is consistent with estimates from both reverberation mapping (3.57[+0.45/-0.37]E7 Msun) and gas kinematics (3.0[+0.75/-2.2]E7 Msun; 1-sigma errors), and our best-fit mass-to-light ratio is consistent with the photometric estimate of M/L(H-band)=0.4+/-0.2 Msun/Lsun. The NIFS kinematics give a central bulge velocity dispersion sigma_c=116+/-3 km/s, bringing this object slightly closer to the M-sigma relation for quiescent galaxies. Although NGC 4151 is one of only a few Seyfert 1 galaxies in which it is possible to obtain a direct dynamical BH mass measurement--and thus, an independent calibration of the reverberation mapping mass scale--the complex bar kinematics makes it less than ideally suited for this purpose. ; Comment: 21 pages, 15 figures. Accepted for publication in ApJ