In late 2014, four images of supernova (SN) "Refsdal," the first known example of a strongly lensed SN with multiple resolved images, were detected in the MACS J1149 galaxy-cluster field. Following the images' discovery, the SN was predicted to reappear within hundreds of days at a new position ~8" away in the field. The observed reappearance in late 2015 makes it possible to carry out Refsdal's original proposal to use a multiply imaged SN to measure the Hubble constant H_0_, since the time delay between appearances should vary inversely with H_0_. Moreover, the position, brightness, and timing of the reappearance enable a novel test of the blind predictions of galaxy-cluster models, which are typically constrained only by the positions of multiply imaged galaxies. We have developed a new photometry pipeline that uses DOLPHOT to measure the fluxes of the five images of SN Refsdal from difference images. We apply four separate techniques to perform a blind measurement of the relative time delays and magnification ratios between the last image SX and the earlier images S1-S4. We measure the relative time delay of SX-S1 to be 376.0_-5.5_^+5.6^days and the relative magnification to be 0.30_-0.3_^+0.5^. This corresponds to a 1.5% precision on the time delay and 17% precision for the magnification ratios and includes uncertainties due to millilensing and microlensing. In an accompanying paper, we place initial and blind constraints on the value of the Hubble constant.