We demonstrate experimentally the existence of magic wavelengths and determine the ratio of oscillator strengths for a single trapped ion. For the first time, two magic wavelengths near 396 nm for the Ca40+ clock transition are measured simultaneously with high precision. By tuning the applied laser to an intermediate wavelength between transitions 4s1/2?4p1/2 and 4s1/2?4p3/2, the sensitivity of the clock transition Stark shift to the oscillator strengths is greatly enhanced. Furthermore, with the measured magic wavelengths, we determine the ratio of the oscillator strengths with a deviation of less than 0.5%. Our experimental method may be applied to measure magic wavelengths for other ion clock transitions. Promisingly, the measurement of these magic wavelengths paves the way to building all-optical trapped ion clocks. � 2015 American Physical Society.