Early in the siege Montgomery identified a hill to the northwest as the key spot from which to threaten the fort. In early October, he “had a road cut to the intended ground and some fascines made.” 
However, his army disapproved of this plan and Montgomery “was informed by Major Brown that a general dissatisfaction prevailed; that unless something was undertaken, in a few days there would be a mutiny.” The army preferred to bombard the fort from afar -- especially the east bank of the Richelieu. Montgomery confessed in a letter to Major-General Philip Schuyler, that when he laid his plans, “I did not consider I was at the head of troops who carry the spirit of freedom into the field, and [who] think for themselves.”
The British took little action to thwart the Americans beyond the exchange of long-range artillery fire. Occasional sorties were made by armed boats, but these efforts ended once a battery was established east of the fort (October 11). On land, a party of Canadians, led by Captains David Monin and Samuel McKay, ambushed some Americans in the woods (October 9), but no sorties were directed against the American camps or gun emplacements.
By the end of October, Montgomery was ready to bring the siege of the fort to a close. He had been reinforced by mortars and ammunition captured at Fort Chambly, and some additional infantry companies from the colonies (specifically, the bulk of the 1st Connecticut and 4th New York regiments). On October 27-28, he abandoned the fortifications south of Fort Saint-Jean and brought his whole force to Grosse Pointe northwest of the fort. There, construction of a new battery overlooking the fort was begun on the evening of October 29.
Lieutenant-Colonel Rudolphus Ritzema of the 1st New York Regiment oversaw the construction:
“In the Evening I was ordered with 200 Men to erect a Battery [the Ground for which having been previously laid out by the Engineers] within 250 Yards of the Fort—In the Morning the Breast Work & Ambresurs completed”
The British soon learned what the Americans were planning:
“Captn Monin and Captn McKay went out this morning in hopes of getting a prisoner, and if possible to survey the Enemys position. An Officer & 25 Men were order’d to be in readiness to support them. They had been out a very little while, when they fell in with a Man who we afterward found was a straggler from a party of 200 Men, who were very near the same Spot [i.e., they captured one of Ritzema’s men]. The Man inform’d us there were 2,000 Men at the rapids (i.e., the lower Camp) and 50 Indians… He shew’d us the place of the Battery…” 
Because of the prisoner’s confession, the new battery quickly became a harrowing place for those that guarded it.
According to Aaron Barlow of the 5th Connecticut Regiment:
“The Regulars discovered our Battery. We guarded it with 100 men, I being one of the Guard. They flung upwards of 100 Bomb shells, some cannon and grape shot at us. Wounded one man, broke two guns. One bomb shell broke within 4 feet of me which made me almost deaf. I believe there were 20 shells broke within 2 rods of me. This night [October 29-30] we dragged four cannon and five mortars to this Breast work in order to play on the Fort.” 
The battery opened on November 1, and was joined by the guns east of the fort. Together, they devastated the fort.
A British officer recorded:
“Large pieces of the Wall were knock’d in. The Chimneys of the House in the South Redoubt were thrown down and the few Corners where some little Shelter from the Weather was to be had were now no longer tenable. A great many shot pass’d thro’ the parapets and some wounded Men behind them. 3 Men were kill’d and 4 or 5 wounded. A good deal of provision was destroy’d.—”
British counterfire was also deadly.
According to Benjamin Trumbull of the 5th Connecticut Regiment:
“On our Side one man was killed right out on the Platform, another had his Leg[,] foot and Thigh torn all to Pieces with a shell, had his Leg cut of[f] about nine o’clock as near the trunk of his body as possible, he bore the Operation with great magnanimity but did not Survive the Night. Three more were wounded but two of them very Slightily.”
Late in the day, Montgomery attempted to open surrender negotiations with the British. An officer in Lamb's artillery recalled:
“I received a message from General Montgomery, ordering me to cease firing till further orders; these orders were extremely disagreeable to me, when I saw some of my men bleeding before my eyes, and dying with the wounds which they had received. On our ceasing to fire, the General ordered a parley to be beat...”
The messenger Montgomery sent was one Lacoste, a Canadian militiaman captured at Longueuil. From him, the British learned that Governor Guy Carleton had been defeated and that there was no hope of relief. The garrison was left with enough rations for 8 days at 2/3 the usual allotment, not including those rations destroyed in the bombardment. The British agreed to a cease fire and pondered their few remaining options.
1. A copy of the letter is available here.
The journal attributed to British Lieutenant John André seemingly confirms the wisdom of Montgomery’s assessment. He noted at the end of the siege:
“We may thank our Enemy in some sort for leaving us in such slight field Works the credit of having been only reduc’d by Famine… Their Batterys might with their numbers [of infantry] by means of Approaches have been brought much closer to our Redouts have overlook’d us, destroyed our breastworks, and by a slaughter from which there cou’d have been no Shelter, have render’d our holding out, a meer sacrifice of Men who might have been reserv’d for better Services.”
2. From a journal attributed to British Lieutenant John André.
3. Sources disagree on the composition of this battery. For example, Montgomery referred to "our battery of four twelve-pounders" (see here). Colonel Timothy Bedel wrote, "I have a battery of four twelve-pounders, one mortar, and three royals, fixing at my post" (see here). Ritzema claimed it consisted of four 12-pounders and six Royal Mortars. An officer with Lamb's Artillery company (likely Captain John Lamb himself) stated that it consisted of "three twelve and one nine-pounders, three mortars, and as many cohorns" (see here). For additional comparison, see this description of the American ordinance used during the siege.