This paper presents the experimental realization of an ultrafast electron microscope operating at a repetition rate of 75 MHz based on a single compact resonant microwave cavity operating in dual mode. This elliptical cavity supports two orthogonal TM$_{110}$ modes with different resonance frequencies that are driven independently. The microwave signals used to drive the two cavity modes are generated from higher harmonics of the same Ti:Sapphire laser oscillator. Therefore the modes are accurately phase-locked, resulting in periodic transverse deflection of electrons described by a Lissajous pattern. By sending the periodically deflected beam through an aperture, ultrashort electron pulses are created at a repetition rate of 75 MHz. Electron pulses with $tau=(750pm10)$ fs pulse duration are created with only $(2.4pm0.1)$ W of microwave input power; with normalized rms emittances of $epsilon_{n,x}=(2.1pm0.2)$ pm rad and $epsilon_{n,y}=(1.3pm0.2)$ pm rad for a peak current of $I_p=(0.4pm0.1)$ nA. This corresponds to an rms normalized peak brightness of $B_{np,textrm{rms}}=(7pm1)times10^6$ A/m$^2$ sr V, equal to previous measurements for the continuous beam. In addition, the FWHM energy spread of $Delta U = (0.90pm0.05)$ eV is also unaffected by the dual mode cavity. This allows for ultrafast pump-probe experiments at the same spatial resolution of the original TEM in which a 75 MHz Ti:Sapphire oscillator can be used for exciting the sample. Moreover, the dual mode cavity can be used as a streak camera or time-of-flight EELS detector with a dynamic range $>10^4$.
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