Following a theoretical proposal [M. Farhat et al., Phys. Rev. Lett. 103, 024301 (2009)], we design, fabricate, and characterize a cloaking structure for elastic waves in 1 mm thin structured polymer plates. The cloak consists of 20 concentric rings of 16 different metamaterials, each being a tailored composite of polyvinyl chloride and polydimethylsiloxane. By using stroboscopic imaging with a camera from the direction normal to the plate, we record movies of the elastic waves for monochromatic plane-wave excitation. We observe good cloaking behavior for carrier frequencies in the range from 200 to 400 Hz (one octave), in good agreement with a complete continuum-mechanics numerical treatment. This system is thus ideally suited for demonstration experiments conveying the ideas of transformation optics.