The firm that was awarded the District's cable television franchise reached an out-of-court settlement of a troublesome lawsuit yesterday, removing the last major obstacle to the city's plans to begin providing cable service by mid-1986.

The agreement, hammered out by lawyers for District Cablevision Inc. (DCI), which was awarded the 15-year franchise, and Capital City Cable, a losing bidder that brought the suit, includes cash payments to Capital City of up to $300,000, according to sources familiar with the negotiations.

Capital City would receive $100,000 within 45 days of U.S. District Court Judge Gerhard A. Gesell's formal dismissal of the antitrust suit. The remainder would be paid by District Cablevision after it has arranged the $50 million in bank financing and $30 million through limited partnerships needed for the project.

District Cablevision also gave Capital City Cable the option to purchase $100,000 worth of stock in the cable television franchise, at $20 a share, which would constitute less than 1.5 percent of the total assets, according to the sources.

"We're very pleased with the settlement and look forward to moving ahead on cable in this city," said Richard Maulsby, executive director of the city's cable television office.

Tyrone Brown, a major shareholder in District Cablevision and a lawyer representing the firm, and Lois Wright, vice president and corporate counsel for Inner City Broadcasting Corp., which has an interest in Capital City Cable, confirmed that a tentative agreement had been reached.

However, the lawyers declined to discuss details until the agreement is signed by all parties involved and formally approved by the court, which probably will occur today.

"All the parties have agreed to the language of the agreement," Brown said. "The parties are treating it the terms as confidential, but DCI considers the settlement as very favorable to it."

The franchise agreement was approved by the D.C. City Council and signed by Mayor Marion Barry late last year. However, until now District Cablevision has refused to ratify the agreement, contending that the suit had had a "chilling" effect on its ability to arrange financing.

The firm has until March 12 to sign the agreement or it will automatically be terminated.

Two other potentially thorny problems were resolved earlier this year.

The Federal Communications Commission granted permission to Chesapeake & Potomac Telephone Co. to install transport lines for city's cable system.

Also, the U.S. Justice Department ruled that the telephone company's involvement in the franchise would not require a waiver from the final court order that broke up the Bell System.

Maulsby said he now expects District Cablevision to sign the franchise agreement "very soon" and arrange for the necessary financing.

A source close to District Cablevision said yesterday, "I think it's safe to say what the company considered to be a major hurdle to financing the franchise has been conquered and the prospects for financing and signing the agreement by March 12 are much improved."

Under the terms of the franchise agreement, District Cablevision has one year to secure financing for construction of the $130 million system. The firm must begin construction of the 1,155-mile cable system within 18 months and must complete construction and provide service to at least 14 percent of the houses and apartments in each of the city's eight wards by April 1986.

By April 1987, cable service must be provided to at least 43 percent of the homes in each of the eight wards.

Capital City, one of three firms that bid on the franchise, originally sought $125.7 million in its lawsuit. The firm alleged that there was a conspiracy, in restraint of trade, to award the franchise to District Cablevision and C&P that would leave the phone company with a monopoly to install cable facilities and to provide cable services in Washington.

Some officials of District Cablevision disputed the allegations and privately criticized Capital City for bringing a "nuisance" suit to try to shake down the District Cablevision.

On Jan. 25, Gesell denied Capital City Cable's request for an injunction and dismissed the District government, City Council members and former city administrator Elijah B. Rogers as defendants in the case.

However, Gesell indicated that the suit would go to trial, but focusing primarily on the question of whether Capital City Cable had been denied due process in bidding on the franchise.

Out-of-court negotiations have been conducted, off and on, for the last month or so, with a breakthrough coming sometime last week, according to sources close to the negotiations.