The multicolour light curves and radial velocities for TYC 1031 1262 1 have been obtained and analysed. TYC 1031 1262 1 includes a Cepheid with a period of 4.15270 +/- 0.00061 d. The orbital period of the system is about 51.2857 +/- 0.0174 d. The pulsation period indicates that the secular period is increasing at a rate of of 2.46 +/- 0.54 min yr(-1). The observed B, V and R magnitudes have been cleaned of the intrinsic variations of the primary star. The remaining light curves that have been obtained consist of eclipses and proximity effects, and these have been analysed to obtain the orbital parameters. The system consists of two evolved stars, F8 II + G6 II, with masses of M-1 = 1.640 +/- 0.151 M-circle dot and M-2 = 0.934 +/- 0.109 M-circle dot and radii of R-1 = 26.9 +/- 0.9 R-circle dot and R-2 = 15.0 +/- 0.7 R-circle dot, respectively. The pulsating star almost fills its corresponding Roche lobe, which indicates the possibility that mass loss or transfer has taken place. We find an average distance of d = 5070 +/- 250 pc, using the BVR and JHK magnitudes, and also the V-band extinction. The kinematic properties and the distance to the Galactic plane (i.e. 970 pc) indicate that it belongs to the thick-disc population. Most of the observed and calculated parameters of TYC 1031 1262 1 lead to its classification as an anomalous Cepheid.