Time series analysis of monthly average total ozone measurements at 37 Dobson stations throughout the world shows that no ozone depletion has occurred at these stations through 1979. The trends at the 14 stations with records from 1958--1979 show an average increase of (1.5¿1.0)% (95% confidence). The trends at the other 23 stations with shorter data records show an average increase of (1.0¿1.7)%; the choice of stations is not critical to the conclusions. The average change is the sum of all long-term global trends. The confidence limits are calculated from the variation among stations and include all effects that differ from station to station. A comprehensive evaluation of model adequacy indicated that variations in the statistical methodology have only small effects on the results. The uncertainty in the observed average change decreases with each year's new data. It is concluded that a downward trend from whatever cause equal to half that hypothesized by current atmospheric models as due to chlorofluorocarbons (CFC's) would be detected. These stations provide a good early warning of any depletion at the measuring stations that might be due to CFC's.