We demonstrate a new method for the direct measurement of atomic dipole transition matrix elements based on techniques developed for quantum information purposes. The scheme consists of measuring dispersive and absorptive off-resonant light-ion interactions and is applicable to many atomic species. We determine the dipole matrix element pertaining to the Ca II H line, i.e. the 4$^2$S$_{1/2} leftrightarrow $ 4$^2$P$_{1/2}$ transition of $^{40}$Ca$^+$, for which we find the value 2.8928(43) ea$_0$. Moreover, the method allows us to deduce the lifetime of the 4$^2$P$_{1/2}$ state to be 6.904(26) ns, which is in agreement with predictions from recent theoretical calculations and resolves a longstanding discrepancy between calculated values and experimental results.
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