The 450 members of the assembly were appointed according to the power-sharing quotas agreed under the Comprehensive Peace Agreement, signed by the Sudanese government and the southern Sudan People's Liberation Movement/Army (SPLM/A) on 9 January.
In a speech to the joint session of the interim National Assembly and the State Council - the two houses of parliament - Bashir affirmed the country's commitment to bringing about lasting development to all parts of the country.
"The setting up of the National Assembly and the State Council is [a] new beginning for national unity and accord," he said.
The president also announced the completion of all political consultations concerning the government of national unity. He is expected to present the country's new leadership in the near future.
Bashir added that the participation of all political forces in the new government would contribute to the implementation of the peace agreement.
During the meeting, Ahmed Ibrahim al-Tahir, the former speaker of the Sudanese parliament and a member of the ruling National Congress Party, was elected speaker. Members of the house elected the SPLM/A delegate, Atim Garang Deng, first deputy speaker.
In accordance with the historical peace agreement, 52 percent of the parliamentary seats were reserved for members of the ruling party, while the SPLM/A was accorded 28 percent.
Northern opposition parties received a further 14 percent, while their counterparts in the south were allotted the remaining six percent of the seats in the new interim assembly.
A number of opposition groups, including the Umma Party of former premier Sadiq al-Mahdi and the Popular Congress Party of Hassan al-Turabi, announced earlier a decision not to participate in the new parliament on the grounds that the president's party unfairly dominated both chambers.
Under the January peace accord, a new legislative chamber will be set up after national elections in three to four years' time, followed by a referendum in which the south will vote on unity or secession.
Meanwhile, discussions were still ongoing with the National Democratic Alliance, the largest coalition of opposition forces, on its participation in the various state institutions.