We present the first results on the history of star formation in the universe based on the "cosmic spectrum," in particular the volume-averaged, luminosity-weighted, stellar absorption-line spectrum of present-day galaxies from the 2dF Galaxy Redshift Survey. This method is novel in that, unlike previous studies, it is not an estimator based on total luminosity density. The cosmic spectrum is fitted with models of population synthesis, tracing the history of star formation before the epoch of the observed galaxies, using a method we have developed that decouples continuum and spectral line variations and is robust against spectrophotometric uncertainties. The cosmic spectrum can only be fitted with models incorporating chemical evolution, and it indicates that there was a peak in the star formation rate (SFR) in the past of at least 3 times the current value and that the increase back to z = 1, assuming it scales as (1 + z)β, has a strong upper limit of β < 5. We find, in the general case, that there is some model degeneracy between star formation at low and high redshift. However, if we incorporate previous work on star formation at z < 1, we can put strong upper limits on the star formation rate at z > 1: e.g., if β > 2, then the SFR for 1 < z < 5 scales as (1 + z)α, with α < 2. This is equivalent to stating that no more than 80% of stars in the universe formed at z > 1. Our results are consistent with the best-fit results from compilations of cosmic SFR estimates based on UV luminosity density, which yield 1.8 < β < 2.9 and - 1.0 < α < 0.7, and are also consistent with estimates of Ω stars based on the K-band luminosity density.