The beta decay of $^{151}$Sm was measured by means of a metallic magnetic calorimeter. The measurement and subsequent analysis yielded a beta spectrum with an outstanding high-energy resolution of about 70 eV (FWHM) at 22 keV and a very low energy threshold well below 400 eV. The spectrum exhibited unexpectedly elevated beta emission probabilities at very low energy that we have not been able to reproduce in our theoretical study. The data analysis was thus scrutinized and an independent analysis of the same data set carried out. All new approaches have confirmed the previously found shape of the beta spectrum. The measured spectrum was compared to predictions from an advanced theoretical modeling that includes the atomic exchange effect, precise radiative corrections as well as the realistic nuclear structure that usually plays an important role in first forbidden non-unique transitions. The measured spectrum was then carefully analyzed to determine the maximum beta energy, which was found to be Q = 76.430(68) keV. The dominant beta decay of $^{151}$Sm populates the ground state of $^{151}$Eu, and a weak beta branch populates the first excited state of $^{151}$Eu. From our measurements, the probabilities of these two branches were determined to be 99.31(11)% and 0.69(11)%, respectively.