By analogy with Earth, methane in the Martian atmosphere is a potential signature of ongoing or past biological activity. During the past decade, Earth-based telescopic observations reported "plumes" of methane of tens of parts per billion by volume (ppbv), and those from Mars orbit showed localized patches, prompting speculation of sources from subsurface bacteria or nonbiological sources. From in situ measurements made with the Tunable Laser Spectrometer (TLS) on Curiosity using a distinctive spectral pattern specific to methane, we report no detection of atmospheric methane with a measured value of 0.18 +/- 0.67 ppbv corresponding to an upper limit of only 1.3 ppbv (95% confidence level), which reduces the probability of current methanogenic microbial activity on Mars and limits the recent contribution from extraplanetary and geologic sources.