September, 1844: Liberty Party Presidential candidate James G. Birney is stricken by a heart attack, causing him to fall from a window into the street, where he is subsequently trampled by a horse. After some wrangling and attempted power plays within the party, Thomas Morris moves from VP nominee to the top spot. Nevertheless, the Liberty Party does worse than it would have, only pulling down 9,500 votes in New York State. Clay wins New York, and the election, with 141 electoral votes to Polk's 134. Seeing the results as a public rejection of the annexation of Texas, President Tyler does not pursue the issue, and does not proceed with his questionable idea of annexation via joint congressional resolution.
In Texas, the election results were seen similarly, but taken much more insultingly. Texas had been attempting to get into the US for some time, facing multiple rejections at the Congressional and now electoral level. Under Henry Clay and with an anti-annexation Senate, prospects didn't look good. Knowing that Texan anger towards the US was at a high point, British ambassador Charles Elliot springs into action. It had long been British policy to try and halt American westward growth, and with the Mexican government frightened by the recent annexation scare, now was the time to strike. Elliot and his French counterpart Alphonse de Saligny negotiate a diplomatic agreement under which Mexico offered to recognize the Republic of Texas, with borders to be determined under Anglo-French mediation.
Texas President Anson Jones, who had been biding his time to see whether annexation or independence would gain more support, feels that public opinion is beginning to turn. He supports the diplomatic agreement, saying that it will only strengthen Texas's hand at the bargaining table with the US, freeing it from troublesome issues of recognition and border disputes. Former President Mirabeau Lamar is strongly in favor of the agreement as well. Ever wary, Jones has the issue put to a national vote. Coasting on anger at the US rejection and impassioned speeches from Lamar, the Texas electorate comes out in favor of the agreement with a slim majority.
Under the British and French mediated settlement, Texas keeps much of their territorial claims. The southern and western boundaries are chipped at, with Texas losing parts of New Mexico and the southern border moved to the Nueces River. The British put pressure on Mexico to allow Texas to keep much of its territory as a buffer against the United States, something which is in both Britain's and Mexico's best interests. Despite the good terms, former President Lamar criticizes the concessions, renewing his old calls for a Texas that stretches to the Pacific, and no doubt laying the groundwork for the Presidential campaign of 1847.
As the expansionist Lamarites plan for a larger Texas, America suddenly finds itself cut off from future growth. Southerners kick and scream as the possibility of Texas annexation, and the possibility of a Mexican War allowing for more territory to be grabbed, evaporates before their eyes. Expansion of slavery to the Midwest and Oregon territories seems unlikely, putting many in fear for the South's Peculiar Institution.
EDIT: I like this, I might actually do a TL myself! Assuming that's not bad manners
EDIT2: \/\/\/\/ I should have known this would already be a popular POD. Poor Birney!