Recently Jamrozy et al. (2009) identified 4C 02.27 (J0935+0204) as the first possible example of a double-double radio source which is optically identified with a quasar (i.e. not a galaxy), at the redshift of z=0.649. The overall projected angular size of the radio source reaches about 1.5, with a prominent core component in the centre. The two opposite pairs of radio lobes might indicate two periods of episodic activity. We report on our short exploratory 1.6-GHz Very Long Baseline Interferometry (VLBI) observations of the innermost radio structure of the quasar, conducted with the electronic European VLBI Network (e-EVN) on 2009 September 30. These revealed a milliarcsecond-scale compact source which is the base of the approaching one of the two symmetric relativistic jets currently supplying the hot spots in the inner pair of the arcsecond-scale radio lobes in 4C 02.27.