At a critical, late stage in the Camp David talks, Egypt's President Anwar Sadat telephoned Jordan's King Hussein. Sadat thought that, for sure, he could talk the king into supporting the next phase of the peace process that would follow an Isaeli-Egyptian peace treaty.

But the connection was so bad that the two men couldn't hear each other -- which is probably just as well. When Hussein heard (in an interview with Barbara Walters) what had been agreed to, he abruptly cancelled a plan to meet Sadat in Morocco right after Camp David.

Since then, the Sadat-Hussein connection, never very cordial on or off the phone, has been out of service. This made the longstanding, close and constructive connection between the United States and Jordan all the more important. But thanks to a combination of tragi-comic miscalculations, mostly on the American side, the Carter-Hussein connection since Camp David hasn't been much better.

All of which is by way of explaining why the Carter administration is attaching extraordinary importance to a visit to Washington this week by King Hussein, even though no big deals will be struck and no hard and fast agreements are likely to be reached. The warmth of the welcome, the pomp and circumstances and the quality as much as the content of the conversation will count for everything.

But the stakes will, nonetheless, be very high.

There are no illusions about winning Hussein over to public support for reviving U.S.-Israeli-Egyptian "autonomy talks" on the future of the Israeli-occupied West Bank and Gaza. But the cooperation of Jordan, as one American diplomat puts it, is "absolutely indispensable to completion of the process at some stage."

So the purpose this week will be the restoration of mutual trust and the easing of Jordanian anxieties. The hope is to reset the stage for the essential business that inevitably will have to be done in concert with Jordan in the months ahead if a resolution of the autonomy issue on paper is actually going to be made to work in practice.

Jordan isn't the only crucial element, of course. The autonomy arrangements for the West Bank and Gaza will also stand or fall depending on the cooperation of the Palestinians, however they may be represented -- by the Palestine Liberation Organization, or by some emerging group of West Bank Arab moderates.

But Jordan has connections, and some influence with Palestinians of all sorts. Moreover, Hussein's government retains a presence, however limited, on the West Bank, an area that, after all, used to be his. Amman still pays some part of the salaries of West Bank officials and dispenses subsidies and loans to West Bank municipalities. It is by no means excluded that a final resolution of the West Bank issue will take the form of its return, by some sort of federation arrangement, to Hussein's dominion.

Hussein also figures increasingly in the interconnected threat to U.S. security in the Persian Gulf. There, Hussein's influences is reinforced by military training programs conducted by the Jordanian armed services. And his policis are influenced, in return, by generous financial aid from the oil producers.

That Hussein cannot be light-heartedly excluded from larger, East-West considerations in the region is pointedly underscored by his scheduled trip to Moscow this summer. Under pressure from his "rejectionist" Arab backers, he may even take that opportunity to buy some Soviet arms.

Which is just one more reason why in the words of one American policymaker, "there is almost universal agreement now on the importance of Jordan."

Why wasn't this obvious all along? The answer is that in some quarters -- and most conspicuously at the diplomatic working level -- it was. Where the point was lost sight of was in the White House. In the Camp David euphoria, Carter either took Hussein for granted or underestimated the depth of the Arab opposition and what this would do to Jordan's relations with its Arab allies.

Whatever the case, Hussein, whose visits to Washington have been an annual routine, was not invited here last year when he was in New York for a U.N. General Assembly meeting and wanted to come. Partly it was presidential pique, officials surmise, and partly it was petty politics. With Carter down in the polls, some advisers didn't want Hussein in town, knocking the president's Camp David success. And then he was invited earlier this year, when Sadat and Israeli Prime Minister Menachem Begin were also booked in, and he didn't want to come, for fear of even appearing to be getting caught up in the Camp David process.

This time around, some quiet, remedial diplomacy has been done in advance. Says one official: "We think we can at least get him to be, if not supportive, at least acquiescing and non-confrontational." Even that much would be a significant and overdue step forward in the precarious Mideast peace process.