The House approved a conference committee report on the District of Columbia's $1.2 billion budget for fiscal 1978 yesterday, but only after narrowly defeating an attempt to kill the proposed convention center at Mount Vernon Square in downtown Washington.

The key vote in an otherwise uneventful debate came on a motion by Rep. Robert E. Bauman (R-Md.) to set aside an amendment paving the way for the $110 million project. Bauman's motion failed, 199 to 190, with one member voting present. Moments lated the House gave conditional approval to the convention center, 199 to 183, again with one member voting present.

An aide to Sen. Patrick Leahy (D-Vt.), chairman of the Senate Appropriations Subcommittee on the District of Columbia, said yesterday that the issue of the convention center is still not settled, although he predicted Senate approval of the conference report.

"Either subcommittee can still kill it," the staff member said, "and we may. If it doesn't live up to the bargain, we'll kill it. No question about it."

The Senate is expected to act on the conference report late this week or early next week.

Under the terms of the Senate and House committees' conference report approved last week, four conditions must be met before work on the convention center can proceed. Rep. William H. Natcher (D-Ky.), chairman of the House Appropriations Subcommittee on the District, told the House that the convention center would not be finally approved until the conditions were met.

The four conditions are a commitment from private businessmen to build millions of dollars worth of private hotels, restaurants and other facilities to serve convention visitors; implementation of a District law to raise tax revenues to pay for land and early operating costs of the center; reexamination of land costs for the center to try to reduce the costs; and a follow-up report that the appropriations D.C. subcommittees of both houses must approve before money can be borrowed for the project.

Natcher pointed out that the District already has enacted two taxes - a 0.9 percent surtax on the city income tax on businesses and an 80-cents-a-night tax on occupancy of hotel rooms - that are expected to raise about $6 million a year. These revenues will be used to pay the cost of land and operating costs of the center when it opens.

Natcher, whose support for the convention center was commended during debate by the District's nonvoting delegate, Walter E. Fauntroy, told the House at one point: "If they want to spend their money for a convention center, every penny of it, in all due respect, they have a right to do it."

Bauman was joined in vocal opposition to the center by Rep. John H. Rousselot (R-Calif.), who recited a series of public works "overruns" in the District - including the Kennedy Center, the National Visitor Center at Union Station and the Metro subway. Natcher, agreeing that Metro would cost $6 billion to $7 billion, noted that neither the Kennedy Center nor the Visitor Center were city projects.

Bauman and Natcher dominated the budget debate, which took about an hour and was attended by about 50 members. Bauman said the arrangement to get nonbinding assurances from locl businesses to invest in supporting facilities was "iffy." Natcher said that "more than the word of some individual" would be needed to proceed with the convention center.

Natcher also said he did not know what procedure should be used by city officials and discounted Bauman's suggestion of embodying promises in legal documents.