Distribution matching is a fixed-length invertible mapping from a uniformly distributed bit sequence to shaped amplitudes and plays an important role in the probabilistic amplitude shaping framework. With conventional constant-composition distribution matching (CCDM), all output sequences have identical composition. In this paper, we propose multiset-partition distribution matching (MPDM), where the composition is constant over all output sequences. When considering the desired distribution as a multiset, MPDM corresponds to partitioning this multiset into equal-sized subsets. We show that MPDM allows addressing more output sequences and, thus, has a lower rate loss than CCDM in all nontrivial cases. By imposing some constraints on the partitioning, a constructive MPDM algorithm is proposed which comprises two parts. A variable-length prefix of the binary data word determines the composition to be used, and the remainder of the input word is mapped with a conventional CCDM algorithm, such as arithmetic coding, according to the chosen composition. Simulations of 64-ary quadrature amplitude modulation over the additive white Gaussian noise channel demonstrate that the block-length saving of MPDM over CCDM for a fixed gap to capacity is approximately a factor of 2.5-5 at medium to high signal-to-noise ratios.