We present 248 precise Doppler measurements of Barnard's Star (Gl 699), the second nearest star system to Earth, obtained from Lick and Keck Observatories during the 25 years between 1987 and 2012. The early precision was 20 m s-1 but was 2 m s-1 during the last 8 years, constituting the most extensive and sensitive search for Doppler signatures of planets around this stellar neighbor. We carefully analyze the 136 Keck radial velocities spanning 8 years by first applying a periodogram analysis to search for nearly circular orbits. We find no significant periodic Doppler signals with amplitudes above ~2 m s-1, setting firm upper limits on the minimum mass (Msin i) of any planets with orbital periods from 0.1 to 1000 days. Using a Monte Carlo analysis for circular orbits, we determine that planetary companions to Barnard's Star with masses above 2 M ⊕ and periods below 10 days would have been detected. Planets with periods up to 2 years and masses above 10 M ⊕ (0.03 M Jup) are also ruled out. A similar analysis allowing for eccentric orbits yields comparable mass limits. The habitable zone of Barnard's Star appears to be devoid of roughly Earth-mass planets or larger, save for face-on orbits. Previous claims of planets around the star by van de Kamp are strongly refuted. The radial velocity of Barnard's Star increases with time at 4.515 ± 0.002 m s-1 yr-1, consistent with the predicted geometrical effect, secular acceleration, that exchanges transverse for radial components of velocity.Based on observations made at Keck Observatory and Lick Observatory.