The frequency noise of a continuous-wave tunable dye laser is suppressed by using the Pound-Drever-Hall frequency locking method. To cope with relatively large high-frequency components in the dye laser frequency noise, a compact intracavity electro-optic modulator that serves as a fast frequency actuator is designed and assembled. The design parameters of the modulator are discussed in detail and the experimental data on the mechanical resonances of two commonly used electro-optic crystals are given. The dye laser is locked to a Fabry-Perot cavity by using the intracavity electro-optic modulator and piezoelectric transducers installed on the cavity mirrors. With only piezoelectric actuators the laser frequency noise is reduced by three orders of magnitude at 100 Hz and the noise is further suppressed by four-fifths from 100 Hz to 50 kHz with a combination of piezoelectric actuators and the intracavity electro-optic modulator. With the suppression of frequency noise implemented, the 2.2 MHz free-running laser linewidth is reduced to 10 kHz.