Over 8 months, we monitored transition frequencies between nearly degenerate, opposite-parity levels in two isotopes of atomic dysprosium (Dy). These frequencies are sensitive to variation of the fine-structure constant (alpha) due to relativistic corrections of opposite sign for the opposite-parity levels. In this unique system, in contrast to atomic-clock comparisons, the difference of the electronic energies of the opposite-parity levels can be monitored directly utilizing a rf electric-dipole transition between them. Our measurements show that the frequency variation of the 3.1-MHz transition in (163)Dy and the 235-MHz transition in (162)Dy are 9.0+/-6.7 Hz/yr and -0.6+/-6.5 Hz/yr, respectively. These results provide a rate of fractional variation of alpha of (-2.7+/-2.6) x 10(-15) yr(-1) (1 sigma) without assumptions on constancy of other fundamental constants, indicating absence of significant variation at the present level of sensitivity.