The expedition, which was placed under the direction of Captain Sturt, of his Majesty's 39th Regiment, commenced its progress down the "Morumbidgee" on the 7th day of January last, having been occupied twenty-one days in performing the journey from Sydney. On the 14th January they entered a new river running from east to west, now called the "Murray," into which the "Morumbidgee" flows. After pursuing the course of the "Murray" for several days, the expedition observed another river (supposed to be that which Captain Sturt discovered on his former expedition), uniting with the "Murray" which they examined about five miles above the junction. The expedition again proceeded down the "Murray," and fell in with another of its tributaries flowing from the south east, which Captain Sturt has designated the "Lindesay;" and on the 8th February the "Murray" was found to enter or form a lake, of from fifty to sixty miles in length, and from thirty to forty in breadth, lying immediately to the eastward of gulf St. Vincent, and extending to the southward, to the shore of "Encounter Bay." Thus has Captain Sturt added largely, and in a highly important degree, to the knowledge previously possessed of the interior. His former expedition ascertained the fate of the rivers Macquarie and Castlereagh, on which occasion he also discovered a river which, there is every reason to believe, is, in ordinary seasons, of considerable magnitude. Should this, as Captain Sturt supposes, prove to be the same river as that above-mentioned, as uniting with the "Murray," the existence of an interior water communication for several hundreds of miles, extending from the northward of "Mount Harris," down to the southern coast of the colony, will have been established. It is to be regretted, that circumstances did not permit of a more perfect examination of the lake, (which has been called "Alexandrina"), as the immediate vicinage of Gulf St. Vincent furnishes a just ground of hope that a more practicable and useful communication may be discovered in that direction, than the channel which leads into "Encounter Bay." The opportunity of recording a second time the services rendered to the colony by Captain Sturt, is as gratifying to the government which directed the undertaking, as it is creditable to the individual who so successfully conducted it to its termination.--It is an additional cause of satisfaction to add, that every one, according to his sphere of action, has a claim to a proportionate degree of applause. All were exposed alike to the same privations and fatigue, and every one submitted with patience, manifesting the most anxious desire for the success of the expedition. The zeal of Mr. George M'Leay, the companion of Captain Sturt, when example was so important, could not fail to have the most salutary effect; and the obedience, steadiness, and good conduct of the men employed, merit the highest praise. By his Excellency's command, ALEXANDER M'LEAY.