Does Dimitri marry Katerina?

Having seduced Katerina—or Katusha, to use the diminutive by which she's known—Prince Dmitri wishes to atone for his sins. When Katusha is sent to a Siberian penal colony after being convicted of murder, Dmitri becomes ever more determined to change, to affect a deep spiritual transformation within his soul. To this end, he tries his best to help Katusha, following her to Siberia and pulling strings to make sure that she's housed with the political prisoners, well away from the serial criminals who try to take advantage of her.
Moreover, Dmitri wants to marry Katusha, but it's all for the wrong reasons. Dmitri doesn't truly love Katusha; he simply wants to cleanse his soul. Marrying Katusha will be a means to a selfish end, nothing more. For her part, Katusha isn't interested. If she's going to marry, then it will be for love. So she declines Dmitri's proposal, opting instead to stay with Valdemar Símonson, a political prisoner whom she meets in Siberia.